Too soft-hearted, apparently

The one species that might have outwitted [the Neanderthals] — or at least proved too difficult to hunt — was the rabbit. As New Scientist reports, John Fa and his research team at the UK's Durrell Wildlife Conservation Trust examined cave sites belonging to both humans and Neanderthals in France and Spain. Judging by animal remains found in the caves, both groups appeared to subsist on large animals like deer until about 30,000 years ago, at which point humans switched over to eating smaller prey like rabbits.

There's no evidence that Neanderthals made a similar move, and that coincides with when their populations went into irrevocable decline. 

(link)