In the later games, 255 is used as the maximum depicted statistical value for an obvious reason. 255 is effective in both mathematics and programming as it forms a complete byte (8 bits: 2^8=256, and we have to allow for zero as a statistical value). By reading a single byte, the game can tell what a character's value is in any given stat. This is why numbers like 65535 (2 bytes: 2^16=65536, with zero as a value) and 16777215 (3 bytes: 2^24=16777216, zero needed) are also sometimes used as caps (the Underwater materia in FFVII, for example, can hold up to 16777215 AP before resetting).

It's also interesting that 255 is represented as FF hexidecimally, although this is simply a noteworthy coincidence.