to elaborate on what max said:
some fish feed on the fins of other fish (can't think of any names just now, but I know they are there)
others are highly territorial- red tail black sharks have been mentioned, and you of course know about bettas- so fin-nipping is a way of defending a territory
others live in groups where establishing a hierarchy is an essential part of group dynamics. With such fish fin-nipping can get more frantic/spiral out of control if there is anything wrong with the group (like not enough members) or surroundings (too small a tank)
many fish will use fin-nipping to attack a sick member of the group; this is a survival thing: let's get rid of this one before he draws the attention of predators to us
a fish that is under stress (poor water stats, illness, pregnancy in livebearers, not enough space etc) may act out of character and fin-nip even if from a normally non-nipping species