"state of inappropriate agitation, fit of childish temper," 1939, American English, of unknown origin. First in Claire Boothe's "Kiss the Boys Good-bye," which gives it a U.S. Southern context. Snitty (adj.) appears to be slightly older.
he was in a snit
sniper
snippers
snippet
snippy
snips
snit
snitch
snite
snivel
snivelling
snob