The story is told of how when Jeff was getting his start, he made a tape of himself doing several of Mel Blanc's characters, including Bugs Bunny. He then took the tape to the producers, and had thus rigged the tape player so that he could use a switch to instantly toggle back and forth between the original Mel recording, and his own recording of the same lines. When he did this, it was impossible for them to tell which was Jeff and which was Mel. Thus his vocal ability was established and his career was launched. The first time he voiced Looney Tunes characters (Bugs Bunny and Daffy Duck, to be more specific) was in Cartoon All-Stars to the Rescue.
One of the more recent times when he voiced several Looney Tunes characters is in the 1991 cartoon short Blooper Bunny: Bugs Bunny's 51st and 1/2 Anniversary Spectacular, in which he voiced Bugs, Daffy, Elmer, and Sam. The cartoon was produced by Greg Ford, a Looney Tunes fan, historian, and consultant. Bergman also was the first one to have the honors of voicing Bugs after Mel Blanc died in 1989. Recently he has become a recurring cast member of Family Guy.