I'm not saying it wasn't the bloodfins (some tetras like to be nippy and i ave no experience with bloodfins), but here's another side to it. Fish often look beaten up a few hours after they die, often because fish will pick on dead fish. So torn fins and stuff can mean they got picked on after death. Look at your other neons, if they have torn fins too.
Otherwise, you can consider some other questions, like water parameters. Ammonia, ph, nitrites, that sort of thing. Also, how long have you had the neons/blood fins? if the neons are new, it can be common for fish to die when you just bring them home (stress of being in the store, transportation, shipping, all that jazz, plus sometimes they can be sick and you might not notice in the store). What other fish are in the tank, and how big is it?