Can MCU memory be subject to bit errors and failures?
yes, they can, but any decent programmer verifies programming.

This is normally done by taking a check-sum of the program. I would actually guess that if it is repeatable, I would reprogram just to be sure, and if it is still repeatable your software is has a small bug. This type of bug can be hell to find.

To check ram, you need to write test characters and read them out, one byte at a type and see if it fails to match.