If you continue running older 32-bit code that hasn't been updated, then, yes, you'd have a time_t value that rolled over. Then again, pretty much any version of Windows or Unix/Linux from prior to the year 2000 w/o the "Y2K" patches would have potentially had similar types of problems post-Y2K.
The point was, if you're running code on an operating system where due diligence has been paid in the application code and updates have been applied to the O.S., then time_t values won't be an issue in the year 2038.