In The Myth of Persecution, Dr. Candida Moss makes several interesting observations, but one that particularly struck home (pp. 103-04):
In a similar way, the author describes religious devotional practices that didn't really take hold until the third century. At the conclusion of the piece, after Polycarp's body is burned for a second time, the Christians steal the fragments of bone and ash that remain and deposit them in an appropriate place for safekeeping. This is not just a concern for proper burial; the author describes Polycarp's remains as "more valuable than precious stones" and says that the remains were placed somewhere that Christians could gather to remember the saints and prepare themselves for their own martyrdom. The situation envisioned here is the veneration of relics.Dr. Moss has a full paper on the dating of "the Martyrdom of Polycarp," which can be accessed for free (at this link). Ultimately, Dr. Moss concludes that the current version of the story was probably composed in the early third century.
... Apart from the Martyrdom of Polycarp, the practice of collecting and venerating the bodies of martyrs is completely unparalleled in the second century. Our next earliest references to relics are from the third century and are much less developed. They may not even be firm references to relics so much as references to the distribution of mementos. In contrast, the Martyrdom of Polycarp does not just refer to relics; it provides an explanation for why the church in Smyrna doesn't have the whole body. That it was necessary to apologize for the absence of relics again presupposes a situation in which relic veneration was already booming. It's difficult to imagine the need to offer this explanation, if the audience wasn't expecting more, and it's difficult to imagine that the audience would have expected more before the third century.
For people looking for examples of the kinds of problems that readers of patristics face, I encourage people to check out Dr. Moss' paper. The work should also help confirm our position on the reliability of the New Testament itself, which is not subject to the same textual transmission difficulties as the story of Polycarp's end.
I think it is worth noting that Dr. Moss dates the work earlier than some of the scholars whose work she is addressing. That said, as Dr. Moss notes in the paper (p. 19): "To my knowledge, no scholar who has regarded MPol as a forgery has ever been convinced that the extant version was written in the middle of the second century."
All the above dove-tails with a point I was noting to someone (in the comment box at GreenBaggins, if I'm not mistaken) that the cult of the dead was not part of the apostolic tradition and only arose later. Even by the time of Augustine in the 4th and 5th centuries, it was not so highly developed as it was in later centuries, such as under King Philip II of Spain (1527-98), who apparently housed 8,000 relics (and over 1,000 paintings) within his palace, el Escorial (see discussion here).