We wouldn't need hostile corroboration to justify an acceptance of the traditional New Testament authorship attributions. The evidence we have from Christian sources is credible. It's early, widespread, and comes from sources who were in a position to know the relevant issues sufficiently well.
Disciples of the apostles lived into the second half of the second century. When Polycarp visited Rome in the middle of the second century to discuss some issues of controversy with the Roman bishop Anicetus, for example, he surely would have interacted with the beliefs of the Roman Christians on issues of New Testament authorship. We have some idea of how those documents were used at that time in Roman church services, for example, from sources like Justin Martyr. Would documents like the gospels and the letters of Paul have been used without any reference to their authorship? When controversial issues arose, like the ones Polycarp discussed with Anicetus, New Testament documents would have been cited in the process. The concept that somebody like Polycarp could live for several decades as a Christian and travel and involve himself in teaching and controversies, as he did, yet have little effect on the authorship attributions of his day, is untenable.
When there were significant disputes over New Testament authorship, such as who wrote Hebrews or whether Peter wrote 2 Peter, those disputes were explicitly and widely acknowledged. Even insignificant disputes, such as those involving the absurd claim that the heretic Cerinthus wrote the gospel of John (a view advocated by only a small minority), left traces in the historical record. For these and other reasons, which I outline here, we have good reason to trust the New Testament authorship attributions of the early Christians.
And just as we should question the motives of the early Christians who commented on such issues, we should question the motives of non-Christian sources as well. It's not as if a Trypho, a Celsus, or a Porphyry is sure to have no bad motives, and to be highly knowledgeable of the subjects he discusses, because he's a non-Christian. We wouldn't conclude that a New Testament authorship attribution must be wrong just because an opponent of Christianity suggested that it was. But the testimony of non-Christian sources is one line of evidence among others. It has some significance.
We know that the early enemies of Christianity were concerned about the subject. The early Christians made much of the significance of eyewitness testimony, for instance, as did the surrounding culture of their day, and Porphyry's efforts in arguing against the traditional authorship attribution of the book of Daniel are a reflection of what we could see with the New Testament documents. But we don't. One of the most significant indications of how the early opponents of Christianity viewed the authorship attributions of the New Testament documents is the lack of interaction with arguments against those attributions. The early Christians widely interacted with Porphyry's arguments against Daniel, and they widely discussed disagreements on issues like who wrote Hebrews and whether the apostle John wrote Revelation. As the historian Paul Maier notes, Dionysius of Alexandria's arguments against Johannine authorship of Revelation are reminiscent of "a good, critical scholar", and the modern scholars who reject Johannine authorship do so "for the very reasons advanced by Dionysius" (Eusebius - The Church History [Grand Rapids, Michigan: Kregel Publications, 1999], p. 285). The people of that time didn't have our level of scholarship, much as we don't have the level of scholarship that will exist a hundred or a thousand years from now, but they weren't so ignorant as to be deceived by a long series of New Testament forgeries that appeared thirty, fifty, or eighty years after the purported authors had died. And if the early Christians had been so undiscerning, at least their enemies would have had reason to exercise more discernment.
Even when a book's authorship attribution was widely accepted, there was an awareness that the attribution could be incorrect. Origen repeatedly affirms the Mosaic authorship of the Pentateuch, yet he's aware of the possibility that Celsus and other critics of Christianity might deny it (Against Celsus, 4:42). When Origen comments elsewhere that the gospels are "unquestionable in the church" (cited in Eusebius, Church History, 6:25:4), he isn't referring to ignorance of other possibilities or a refusal to consider counterarguments. He's referring to a consensus that was reached by a community of people who were concerned with evidence and had repeatedly shown a willingness to question other books.
Augustine refers to how critics of the gospels in his day would "assert that the disciples claimed more [in the gospels] for their Master than He really was" (The Harmony Of The Gospels, 1:7:11; see also 1:16:24), and would object that Jesus didn't leave any writings Himself, but there seems to have been widespread acceptance of the attribution of the gospels to Jesus' disciples (Matthew and John) and their followers (Mark and Luke). These critics weren't objecting to the authorship attributions, but were objecting on other grounds instead. Other sources suggest that Augustine's assessment was accurate.
Below are some comments Charles McIlvaine wrote on this subject in the nineteenth century. His focus is on the gospels and Acts, not the entirety of the New Testament, and I don't agree with every comment he makes. He's discussing enemies of Christianity such as Celsus and Porphyry, so he doesn't mention some less hostile sources who could be included. For example, heretical groups like the Ebionites often accepted the authorship attributions of New Testament documents they opposed, even though arguing against those attributions, if that seemed plausible to them, would have been more effective. The historian Philip Schaff gives another example:
"These heretical testimonies [in support of the fourth gospel] are almost decisive by themselves. The Gnostics would rather have rejected the fourth Gospel altogether, as Marcion actually did, from doctrinal objection. They certainly would not have received it from the Catholic church, as little as the church would have received it from the Gnostics. The concurrent reception of the Gospel by both at so early a date is conclusive evidence of its genuineness. 'The Gnostics of that date,' says Dr. Abbot, 'received it because they could not help it. They would not have admitted the authority of a book which could be reconciled with their doctrines only by the most forced interpretation, if they could have destroyed its authority by denying its genuineness. Its genuineness could then be easily ascertained. Ephesus was one of the principal cities of the Eastern world, the centre of extensive commerce, the metropolis of Asia Minor. Hundreds, if not thousands, of people were living who had known the apostle John. The question whether he, the beloved disciple, had committed to writing his recollections of his Master’s life and teaching, was one of the greatest interest. The fact of the reception of the fourth Gospel as his work at so early a date, by parties so violently opposed to each other, proves that the evidence of its genuineness was decisive. This argument is further confirmed by the use of the Gospel by the opposing parties in the later Montanistic controversy, and in the disputes about the time of celebrating Easter.'" (History Of The Christian Church, 1:12:83)
Charles McIlvaine's focus, below, is on the most hostile enemies of Christianity, not individuals who considered themselves Christian. But we should keep groups like the Ebionites and the Gnostics in mind, since much the same can be said about them.
I've read all of the extant fragments of Celsus, but I haven't read all of the relevant fragments by or about the other sources McIlvaine discusses. (For those who have read some of these fragments in the works of R. Joseph Hoffmann, see here for a discussion of some of the problems with relying on his material.) I can't affirm everything McIlvaine writes below. I'm quoting him because he's generally credible, he makes some good points, and I agree with the general thrust of his comments. He writes:
It may be said, with some appearance of a plausible objection to the testimony hitherto produced, that it is all derived, either from the devoted friends of the gospel, or else from those who professed to be its disciples. Is there no testimony from enemies? The books of the New Testament were widely circulated; christian advocates, in their controversies with the Heathen, freely appealed to them; Heathens, in their works of attack and defence, must have spoken of them. In what light did they regard them? Did they ascribe them to their reputed authors, or question their authenticity? Now we do not grant that the testimony already produced is justly liable to the least disparagement on account of its having been derived exclusively from the friends of Christ. That certain ancients believed the facts contained in Caesar's Commentaries has never been supposed to diminish the value of their testimony to the authenticity of that work. We will take occasion, by and by, to show that the very fact that an early witness to the New Testament history was not an enemy, but a friend, of the gospel, and had become a friend from having been once an enemy, is just the ingredient in his testimony that gives it peculiar conclusiveness. Still, however, we are under no temptation to undervalue the importance of an appeal to the opinions of adversaries. Let us inquire of enemies as well as friends -- and first of Julian.
Julian, the emperor, united intelligence, learning, and power, with a persecuting zeal, in a resolute effort to root out christianity. In the year 361, he composed a work against its claims. We may be well assured that if any thing could have been said against the authenticity of its books, he would have used it. His work is not extant; but from long extracts, found in the answer by Cyril, a few tears after, as well as from the statements of his opinions and arguments by this writer, it is unquestionable that Julian bore witness to the authenticity of the four Gospels and of the Acts of the Apostles. He concedes, and argues from, their early date; quotes them by name as the genuine works of their reputed authors; proceeds upon the supposition, as a thing undeniable, that they were the only historical books which Christians received as canonical -- the only authentic narratives of Christ and his apostles, and of the doctrine they delivered. He has also quoted, or plainly referred to, the Epistles to the Romans, Corinthians, and Galatians, and nowhere insinuates that the authenticity of any portion of the New Testament could reasonably be questioned. Let us ascend a little higher.
Hierocles, president of Bithynia, and a learned man, of about the year 303, united, with a cruel persecution of Christians, the publication of a book against christianity, in which, instead of issuing even the least suspicion that the New Testament was not written by those to whom its several parts were ascribed, he confines his effort to the hunt of internal flaws and contradictions. Besides this tacit acknowledgment, his work, or the extracts of it that remain, refer to, at least, six out of the eight writers of the books of the New Testament. Let us ascend still higher.
Porphyry, universally allowed to have been the most severe and formidable adversary, in all primitive antiquity, wrote, about the year 270, a work against christianity. It is evident that he was well acquainted with the New Testament. In the little that has been preserved of his writings, there are plain references to the Gospels of Matthew, Mark, and John, the Acts of the Apostles, and the Epistle to the Galatians. Speaking of Christians, he calls Matthew their evangelist. "He possessed every advantage which natural abilities or political situation could afford, to discover whether the New Testament was a genuine work of the apostles and evangelists, or whether it was imposed upon the world after the decease of its pretended authors. But no trace of this suspicion is any where to be found; nor did it ever occur to Porphyry to suppose that it was spurious." How well this ingenious writer understood the value of an argument against the authenticity of a book of scripture, and how greedily he would have enlisted it in his war against christianity, could he have found such a weapon, is evident from his well known effort to escape the prophetic inspiration of the book of Daniel, by denying that it was written in the times of that prophet. We may ascend still higher.
Celsus, esteemed a man of learning among the ancients, and a wonderful philosopher among modern infidels, wrote a laboured argument against the Christians. He flourished in the year 176, or about seventy-six years after the death of St. John. None can accuse him of a want of zeal to ruin christianity. None can complain against his testimony, as deficient in antiquity. An industrious, ingenious, learned, adversary of that age, must have known whatever was suspicious in the authorship of the New Testament writings. His book entitled "The True Word," is unhappily lost, but in the answer, composed by Origen, the extracts from it are so large that it is difficult to find of any ancient book, not extant, more extensive remains. The author quotes, from the Gospels, such a variety of particulars, even in these fragments, that the enumeration would prove almost an abridgement of the Gospel narrative. Origen has noticed in them about eighty quotations from the books of the New Testament, or references to them. Among these there is abundant evidence that Celsus was acquainted with the Gospels of Matthew, Luke and John. Several of Paul's Epistles are alluded to. His whole argument proceeds upon the concession that the christian scriptures were the works of the authors to whom they were ascribed. Such a thing as a suspicion, to the contrary, is not breathed; and yet no man ever wrote against christianity with greater virulence. Hence it appears, "by the testimony of one of the most malicious adversaries the christian religion ever had, and who was also a man of considerable parts and learning, that the writings of the evangelists were extant in his time, which was the next century to that in which the apostles lived; and that those accounts were written by Christ's own disciples, and, consequently, in the very age in which the facts there related, were done, and when, therefore, it would have been the easiest thing in the world to have convicted them of falsehood, if they had not been true." "Who can forbear (says the devout Doddridge) adoring the depth of divine wisdom, in laying up such a firm foundation of our faith in the gospel history, in the writings of one who was so inveterate an enemy to it, and so indefatigable in his attempts to overthrow it." Who, I will add, can help the acknowledgment that in Celsus, Porphyry, Hierocles, and Julian, all of them learned controversialists, as well as devoted opponents and persecutors of Christians, extending their testimony, from the seventieth year after the last of the apostles, to the year of our Lord 361 -- every reasonable demand for the testimony of enemies is fully met, and a gracious Providence has perfected the external evidence for the authenticity of the New Testament?