I think it doesn't necessarily mean they all made like rabbits and vanished - it alludes to the expression, quoted by another poster - "ich heisse Hase, ich weiss von nix". So the moment I turn up somewhere, everybody is named Hase, i.e. "nobody knows nuttin' ". Hence all the excellent suggestions in that vein are the ones you should choose from. (You can't very well tell me "My name's Hase, I don't know nuttin'" if you've headed for the hills when you saw me coming.)
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In fact, the "ich heisse Hase" bit is from some sort of anecdote or urban legend, or maybe a movie scene, where the guy's name actually is Hase, or he says it is. At any rate there's a narrative of some sort behind the expression; it's not meant to allude specifically to rabbits. An old German told it to me fairly recently and that's how I know.
peer agreement (net): +5