Tom has a book called "How To Catch a Mouse," which includes the dictum: "A cornered mouse NEVER FIGHTS." After cornering Jerry, he leaps off screen, where sounds and cartoon effects indicate a big fight. Seconds later he re-enters, with a black eye and a lump on his head, and turns to the camera and says, as if from an echo chamber, "Don't you believe it!"
This was clearly a well-known catchphrase in the 1940s. But from what? This film may provide a clue:
Don't You Believe It (1943) (1943). Did the narrator, John Nesbitt, use this phrase in the same echoing voice? Were there several "Passing Parade" shorts, like this one, that were devoted to debunking popular falsities?
Or did the catchphrase come from radio? A show called "Don't You Believe It" ran from 1938-1947.
Anyone who knows this answer for certain is invited to edit this entry. Note that Tom uses the phrase again, more memorably, in
The Missing Mouse (1953) (1953). And Bugs Bunny says it in
Big Top Bunny (1951) (1951).
See: a discussion of this question on Google
See: another discussion on TV.com