OK, pain is definitely the culprit. I just curbstomped (fairly literally) a half-dozen badgers with an elephant immune to pain, and only occasionally had tiredness become a problem (and simply walking around for a little while cured that right up).
This means that the nibble attacks are causing constant fainting from an elephant (mass: 5,000 kg) being bitten about 30 times from badgers or other small mammals (I got this to happen with hoary marmots and rabbits, as well). The nibbles only seem to need to be able to break skin to cause enough pain to overwhelm the elephant.
I'm guessing this is a problem of how pain works in general - it may be that if 1 pain is generated for breaking the skin on a hamster with a sword and 1 pain is generated for breaking the skin of an elephant with a needle - all pain is tallied up the same. Once 30 pain points from 30 individual 1 pain injuries is added up, it causes fainting, wheras having a shattered bone would generate about 20 pain. (And creature size isn't a factor.)
Simply using low-integer values may be the problem here.
Likewise, it takes absurdly long for the elephant to actually die, because the low integer value for bleeding and simple size of the elephant means that the only way for a badger to actually inflict bleeding (the only way to kill the elephant) is to attack the softer/smaller trunk, ear, or tail parts of the elephant - repeatedly gnawing at the eyes and throat do essentially no damage (although eyes lose function and both eyes and throat are "dented", they never start bleeding).
There is one open issue that at least partially matches what I am talking about -
http://www.bay12games.com/dwarves/mantisbt/view.php?id=2115 I'm not sure if this really is "the same issue", however, since he's talking about fainting from a broken bone, and I'm talking about fainting in a couple turns from nibbles.
EDIT:
To be more specific, I have had elephants be defeated by as few as 3 hoary marmots, thanks entirely to fainting from the pain.
On the front of the other issue I was talking about before this one, I'm fairly sure the problem with badgers being too easily enraged (and I didn't see any bugs on Mantis for this one) is caused by the dwarves and livestock basically not giving badgers any space until they actually do flip out.
Essentially, badgers path towards the yak I got from the embark that's minding its own business, grazing. The badgers see the yak, and flee in terror from the larger creature. Then they path back to the yak. Then they flee in terror. Then they path back to the yak. Then they see the yak and get enraged for being "disturbed" so many times, and attack the yak, who suddenly flees in terror from the suddenly hostile badgers. When the yak runs away, the badger decides to instead lock onto some passing herbalist, instead. The badger then chases the herbalist all the way around the map.
Any time the badger falls out of rage, the dwarves start ignoring the badger again, and try to path past the badger back to doing whatever it was they were doing beforehand. When they get close to the badger, it suddenly gets pissed again, and flies back into a rage, and chases the herbalist around some more.
The problem is, in a nutshell, that there's no real way to tell your dwarves or animals to leave badgers alone the way that they would avoid an animal that is dangerous all the time, like a hippo, so they keep wandering up to the badgers and enraging them. This means badgers attack far more often than fiercer creatures that most dwarves just leave well enough alone.
Likewise, badgers just keep trying to path into places occupied by creatures that are bigger than them, and that they are afraid of, even though they are scared off the first half-dozen times. After being scared off a couple times, maybe they should just try to find some other spot to mill around? It hardly counts as "defending your territory" if you're invading someone else's turf.
I don't see any issue like this posted on Mantis searching for "rage" and "badger". I guess this qualifies more as a bug (or rather, the lack of a feature in AI that should be present) than the previous, since this isn't specifically a balance issue. (Of course, maybe flying into a rage just because someone comes too close to you and attacking other creatures when the setting is only PRONE_TO_RAGE:1, the lowest possible setting, is a balance issue itself.)