NOTE: I'm not a biologist, nutritionist or a biochemist. If you are, please consider fact-checking me here.
Humans are omnivores. We hunt and gather. Many of us like tasty meat. Rabbit, in the wild, turns out to be a nice sort of meat to nab and chomp because they're not too bright (easily trapped), have useful pelts (warm!) and consist of very lean meat. I used to think they were fatty, but I was quite wrong. It's the domestic bunnies that are fat because they don't get exercise. Wild rabbits, on the other hand, spend most of their short lives running away from things and doing other good cardio exercises like breathing really really fast (watch their noses). So they're quite lean. Also, it's extremely hard to 'hunt an area out' of rabbits, because they breed so gosh darned quick. They're nature's Dorito - 'crunch all you want, they'll make more!'
The problem comes when a hunter in hard times comes to depend solely or heavily on rabbit meat. The problem is that rabbit meat is almost entirely protein. Wait! I hear you say. Isn't protein good for you? Well, yes, it is, especially if you're doing physical work (like surviving in the woods) because it allows you to replace damaged muscle tissue and the like. It's built of amino acids. The problem is that there's almost no carbohydrates in rabbit, and almost no fat. This is a problem because the only thing our bodies can 'burn' for fuel are sugars - essentially glucose. This is why, when you're put on fluids in a hospital, you're essentially taking in sugar water - the 'calorie' in Isocal HN is sugar. So if you're living on a diet of rabbit, you're getting a lot of protein, and a bunch of indigestible bits (solid waste).
Now, the human body can metabolize proteins into sugars, and thence into fuel - the problem is that not only is it an energy-intensive process, but it's also limited by liver capacity. In protein catabolism, the human body breaks proteins down into amino acids. Then, some of these are used to make body proteins (muscle tissue, for example); others are converted to various compounds containing nitrogen the body needs, and the rest are converted to carbon skeletons. To do this, the body strips the ammonia out of the glutamate amino acid, which results in NH4. Since ammonia is toxic, it is converted into urea which can be eliminated since it's water soluble. Thank your kidneys next time you eliminate some.
At this point, the various carbon skeletons can be broken into pyruvates, which in turn can become either ATP (yay!) or glucose (yay!). Whew.
The problem is that this process uses up an enormous amount of ATP, which is used to convert NH4 into urea as well as to break down the carbon skeletons and create glycogen out of glucose. As a result, it's not very energy efficient to metabolize protein, although it is net positive. The ATP has to be replaced! How? Well, your liver does that. But in order to make ATP, it needs to use oxygen. So the limiting factor, it turns out, is the amount of oxygen that your liver can grab and process per unit time (say, a day). In the course of a day, the liver using protein inputs alone can (net) produce enough ATP to result in around 400 grams of glucose.
That's not a lot. That's approximately 1600 kilocalories. 1600 kcal/day is the metabolic rate of a bedridden patient, not someone running around doing active stuff.
So the body needs more fuel, even though it's taking in loads of protein. The nice thing about protein is that raw amino acids can be flushed out of the blood by the kidneys also. So long as that N molecule doesn't leave the amino acid, it's safe. So any amino acids the body doesn't have the resources to process get flushed (haha) out of the system, no harm, no foul. This isn't true in the extremes; there are a variety of hypotheses about the damage which excessive protein intake (and intake of 'deranged' proteins) can do to the body, ranging from cancer to diabetes to what-have-you.
And there you have it. The body, no matter how much protein you feed it, can only produce around two-thirds to half of a 'normal' metabolic supply of glucose. What happens then?
In most people, especially us fatty Americans, the body starts to burn fat. Fat is much much easier to crack to glucose than protein, and if the body has any fat in storage, it'll start raiding that supply. This, of course, is why high protein diets work so well - and why they go so wrong if you cheat with carbs at all. But if you're our desperate hunter in the wild and all you can get is rabbit, why, you're going to burn through all your fat. You'll do it a lot faster if it's cold out and you're keeping yourself warm as well - which makes it harder to stay warm, since fat is insulation.
So finally, this explains how people can starve to death while cramming their bellies full of wild rabbit meat. It's all protein, and no matter how much they distend their tummy on it, the body has a hard limit on how much it can process.
In the wild, people in this state have been observed to become starved for fat. Our wiser predecessors used to make pemmican as a travel food. Pemmican was 20 percent protein - and around 80 % fat. Guess why?
Rabbit starvation. If you find yourself on your own in the wild, remember that while roast rabbit is tasty, you should find yourself a nice fatty fish or coon once in a while. Or, as BlackPawn suggests, maybe a koala or a wombat. Mmm. Glowing Fish reminds us that hawthorne berries are a lovely bland source of carbohydrates, and Orange Julius chimes in to suggest any budding survivalists avoid early-season (spring, early summer) rabbits since they're full of parasites. The later seasons rabbits who survive have flushed most of these nasties out.
Sources:
MedBio, the U.S. NIH's online references
University of South Australia's online references. Australia's got a lot of rabbits, yep yep.
The Primal Feast, by Susan Allgood, on Google Books.