At last we come to the epistles of John, and thus the end of Robinson's treatment of the dates of the NT. There remains to consider his "Post-Apostolic Postscript," which deals with select texts from the Apostolic Fathers (and which I aim to treat in several posts, each dealing with a different text). The epistles of John are particularly resistant to chronological situation. Even 2 Peter, which is notoriously resistant, at least lets us know that it most postdate at least two Pauline epistles (cf. 3:15-16). There is nothing remotely comparable in the Johannine epistles. Thus does Robinson do what I think to be just about the only viable procedure: he places them at the end of his investigation, and with a framework already in place asks where they best fit. He argues, reasonably, that they probably were written in parallel with John's Gospel: neither before nor after that gospel, but rather as the gospel was coming together. He argues more specifically that they postdate what we might call a "Proto-John," i.e. absent the Prologue (1:1-18) or the Epilogue (ch. 21), but predate the completed gospel; thus he situates them at c. 60-65, just prior to when he dates the completed Gospel of John.
The basic idea that the tradition that eventuated in the Fourth Gospel developed over some time seems reasonable, as too is the hypothesis that while this development was yet ongoing letters were produced by the same circle or author. As such, I can hardly rule out that in 1, 2, and 3 John we have one or more Johannine letters produced in chronological parallel with the gospel. I am a less-than-convinced that there was a Proto-John, as Robinson proposes. Really, the only evidence that John ever existed absent these passages is that it could have existed absent these passages; in other words, the hypothesis becomes its own confirmation, a dubious way to proceed. More generally, we really don't know that John's Gospel went through various editorial stages, as the Bultmannian and neo-Bultmannian traditions have tended to suppose; such supposition tended to be programmatic rather than demonstrated. At the very least, Robinson seems to overstate his case here, as he does with other parts of his hypothesis. For instance, yes, 1 John 1:1-4 could be read as a "first draft" of John 1:1-18, as Robinson argues; but it could also at least as plausibly be read as correcting a docetic misreading of the Gospel (as, for instance, Raymond Brown argued). All this however speaks ultimately to the difficulty of situating the letters in time.
My own feeling is that the particularly close literary relationship between the Gospel of John and 1 John place them within proximity to each other. My thinking is that 1 John represents more fully the theology behind the gospel, and that the time when the author would be most disposed to explicate said theology most fully would be around the time that the gospel circulated: either as he was working on the gospel, or to amplify it afterwards. Thus, granting that the Gospel is to be dated around 65, I would think that the mid-60s is best for 1 John. Frankly, at this point I remain agnostic on II and III John.