I can't find much more info about this 2034 date than rote repetition. The only site I can find that says anything more says:

2034-09-30 Sat - Time overflow in some ancient UNIX, which was fixed long ago. Genuine, but no longer significant. See 2038.

Wait. Here's one that says:

September 30, 2034 - UNIX time function overflow per USAF website (May be one of those nasty site-specific dates)

I think that we can both agree that the problem in 2038 is that a signed 32-bit integer will no longer be able to contain the number of seconds after 1969, right? Well, my Solaris man page for time(2) says:

The time() function returns the value of time in seconds since 00:00:00 UTC, January 1, 1970.

which seems like the same description to me.