they won't kill themselves from fin/tail biting. They might want to make you kill them, or yourself, but nothing so drastic. Clean water just treats the symptom, to find the cause, you have to look at his environment. Did you add something new? Stress is one of the causes of fin biting. The other is the opposite, sort of. Boredom. Is there a hiding spot in his tank/bowl? Does he have nice plants to swim in or rest on? Do you have any interaction with him? Show him a mirror to flare at, etc.
And the final is the one we can't do anything about. They just do it. No reason. They want shorter fins, they have an itch, hey it's a new style, they -can-, or just because it will annoy the crap outta you. All reasons for tail biting.