To fill up what's lacking in Christ's afflictions

Now I rejoice in my sufferings for your sake, and in my flesh I am filling up what is lacking in Christ's afflictions for the sake of his body, that is, the church (Col 1:24).

1. This is a somewhat puzzling passage. It's not entirely unexampled in Paul's writings. 2 Cor 1:5 has a similar sentiment ("For as we share abundantly in Christ's sufferings").

2. In terms of historical theology, this is a traditional prooftext for the Catholic treasury of merit:

1475 In the communion of saints, "a perennial link of charity exists between the faithful who have already reached their heavenly home, those who are expiating their sins in purgatory and those who are still pilgrims on earth. between them there is, too, an abundant exchange of all good things."86 In this wonderful exchange, the holiness of one profits others, well beyond the harm that the sin of one could cause others. Thus recourse to the communion of saints lets the contrite sinner be more promptly and efficaciously purified of the punishments for sin.

1476 We also call these spiritual goods of the communion of saints the Church's treasury, which is "not the sum total of the material goods which have accumulated during the course of the centuries. On the contrary the 'treasury of the Church' is the infinite value, which can never be exhausted, which Christ's merits have before God. They were offered so that the whole of mankind could be set free from sin and attain communion with the Father. In Christ, the Redeemer himself, the satisfactions and merits of his Redemption exist and find their effficacy."87

1477 "This treasury includes as well the prayers and good works of the Blessed Virgin Mary. They are truly immense, unfathomable, and even pristine in their value before God. In the treasury, too, are the prayers and good works of all the saints, all those who have followed in the footsteps of Christ the Lord and by his grace have made their lives holy and carried out the mission the Father entrusted to them. In this way they attained their own salvation and at the same time cooperated in saving their brothers in the unity of the Mystical Body."88

1478 An indulgence is obtained through the Church who, by virtue of the power of binding and loosing granted her by Christ Jesus, intervenes in favor of individual Christians and opens for them the treasury of the merits of Christ and the saints to obtain from the Father of mercies the remission of the temporal punishments due for their sins. Thus the Church does not want simply to come to the aid of these Christians, but also to spur them to works of devotion, penance, and charity.89

http://www.vatican.va/archive/ENG0015/_P4G.HTM

But even if, for argument's sake, Col 1:24 was consistent with the treasury of merit, it can't be used to prove that dogma inasmuch as the treasure of merit is a much more specific and complex notion than the content of Col 1:24. Assuming that Col 1:24 is consistent with the verity of the Thesaurus Meritorum, it's equally consistent with the falsity of the Thesaurus Meritorum.

3. It can't mean that Christian suffering supplements the atonement of Christ. For one thing, in this very same letter, Paul stresses the sufficiency of the atonement: 

13 And you, who were dead in your trespasses and the uncircumcision of your flesh, God made alive together with him, having forgiven us all our trespasses, 14 by canceling the record of debt that stood against us with its legal demands. This he set aside, nailing it to the cross (Col 2:13-14). 

Moreover, there's nothing specifically redemptive about Paul's terminology in Col 1:24. He doesn't refer to the "blood", "cross", or "death" of Christ. 

4. Paul's statement probably includes the affliction of Christian persecution. But the language is more generic.

5. To some degree, Christian suffering poses a paradox. Sometimes the most devout suffer the most. Why is that?

In the case of persecution, the rationale is obvious. A public witness to an unbelieving world.

But much Christian suffering isn't directly testimonial in that respect. Some Christian suffering appears to be pointless, or directed at those who leave deserve it. Why do some of the most saintly believers endure the most? 

Perhaps Col 1:24 provides an answer, or at least a hint. Imagine if Christians were exempt from suffering. It would be much harder for us to appreciate what Jesus went through on our behalf. His suffering would be an intellectual abstraction rather than an emotional experience we can relate to. But because Christ and Christians share suffering in common, that makes the suffering of Christ more real to us.

Jesus didn't merely suffer redemptively. Consider his reaction to the death of Lazarus. With that in mind, consider the bereavement he must have undergone when beloved relatives died. He probably outlived his stepdad. May well have outlived his grandparents. Maybe an aunt and uncle. Like you and me, he knew what it's like to bury a loved one. He knew the grieving heart firsthand.

In addition, his own family misunderstood him. There was no one he could relate to on his own level.

Christians participate in the afflictions of Christ in the sense that suffering is the common lot of man, And Jesus entered into our sorrowful experience. To that degree, it's reciprocal.