To further expand on something hinted at a couple of times in my previous post, history and Tradition generally trace connection to the original Apostles more through succession to a particular See than through the actual line of ordination.
It’s true that (by Tradition and some pretty good historical likelihood, though no rigorous proof) all bishops are consecrated by other valid bishops, who were granted their office in the same way all the way back to the Church’s original bishops, the Apostles. That is indeed a Catholic (and Orthodox, and probably Anglican) claim.
However, the papal succession, and that of most other Apostolic Sees, does not rest on who ordained whom, but on the antiquity and traditional prerogatives of the See.
The current Patriarch of Antioch, for example (and there may be more than one who claims the title, with both Orthodox and Catholic lines out there) is no doubt proud to have succeeded to an episcopate once held by Peter, even if his own line of ordination actually would trace to a different apostle.
Likewise, the various Mar Thoma bishops in India undoubtedly regard themselves as “successors of St. Thomas” in that they currently lead the community he founded, whatever their actual lineage. (Though, assuming Thomas really did bring Christianity to India, it’s likely that they are also his successors by line of ordination.)
Finally, the papacy is tied to the See of Rome, not to a particular line of ordination. As far as I know, Karol Wojtyla was not among the bishops who consecrated Joseph Ratzinger a bishop, nor did John Paul II get to choose the next Pope – yet Benedict XVI is his lawful successor nonetheless, and both are lawful successors of Peter as bishop of Rome. That’s true even if their actual lines of ordination could be traced back through history and were found to originate with different Apostles entirely.