A diet of rabbit will make you sick after about a week.
Rabbit eaters, if they have no fat from another source—beaver, moose, fish—will develop diarrhea in about a week, with headache, lassitude and vague discomfort. If there are enough rabbits, the people eat till their stomachs are distended; but no matter how much they eat they feel unsatisfied.
https://en.wikipedia.org/wiki/Protein_poisoning