Not necessarily. The teacher might have thought that you were very good, but not good enough. But that doesn't mean you don't try and give up hope!!!
Different jazz band leaders/instructors will have different requirements, but you can bet any jazz band audition will have these two: Demonstrating good sight reading ability, and demonstrating good ability to improvise - especially the latter. Jazz is all about the ability to improvise.
Clarinets have a long tradition in jazz -- check out Benny Goodman. I would suggest that a few clarinet players who are interested in Jazz Band get together, learn a couple of pieces, and ask the band teacher to listen.
is a cornet used in a jazz band
Hatano Jazz Band was created in 1912.
original dixieland jazz band
The Rhythm Section Jazz Band was created in 2002.
Good Reviews - Louis Armstrong Band
The Original Salty Dogs Jazz Band was created in 1947.
Is a great Jazz Band
Jazz Blue Swings Vocalist - she had a jazz band.
As far as I know, I haven't seen a single clarinet in a jazz band (sure in a marching band, but not in jazz). My friends who play clarinet are also in the jazz band, but intead of playing their clarinets, their switch/use saxophones instead. So, no, I don't think clarinet is a jazz instrument.
Lisa's teacher is Miss Hoover (Elizabeth). She was replaced for the episode 'Lisa's Substitute' by substitute teacher, Mr Bergstrom. her specific jazz teacher was Bleeding Gums Murphy