I am really shocked that a great director like Chuck Jones started out making some of the most incredibly boring cartoons I've ever seen. I did not laugh once throughout this short, and it's a Bugs Bunny cartoon, for Christ's sake! Bugs Bunny cartoons are always funny, not boring! Alas, this short turns out to be Good Night Elmer (another incredibly boring Jones short) with the addition of Bugs Bunny.
The first warning sign of a dull cartoon is always no gag payoff. Good Night Elmer was boring because it dragged on the same two gags forever with predictable payoff. This cartoon, on the other hand, is afflicted with the second warning sign of a dull cartoon: there's too much dialogue. The cartoon at least has more than two gags up its sleeve, but most of them seem longer than they are thanks to the immense padding of the dialogue. At one point, Elmer finishes eating dinner, and comments, "That was weawwy awfuwwy good weg of wamb," possibly the most redundant dialogue I've ever heard in a cartoon (characters reading text out loud in the later-era Woody Woodpecker cartoons doesn't count in my book). Even though this cartoon is only 8 minutes long, it feels like 20 thanks to redundant dialogue like this.
Elmer's Pet Rabbit was not a fun cartoon for me, but if you've sold your soul to Chuck Jones and are unable to acknowledge that he directed a few clunkers during his career, you might enjoy it.