At the precise date of the epochalypse, the date will roll over for the current time_t.

But one second before that, it will blow up for a system that peeks 1 second into the future (e.g a control system).

One day before, the date will be screwed up for any program doing addition with one day such as a rental system.

In the years before 2038, systems that do calculations on year scales will fail, like perhaps insurance or mortgage and similar.