False.
A Game Boy game's internal battery has a max out limit of 256, and can never be reached. Therefore, the highest number possible on the Play Time on a Generation I and II game is 255:59.
The internal battery was there to power the internal clock even when the game wasn't running.
What caused the clock to stop at 255 is the way they saved variables before you had more than 2 MB space on your cartridge: In bytes. A byte can go from 0 to 255, which includes all possible ways to arrange 8 Bits.