Can rabbits get diabetes?

The reason I ask is because Sadie has suddenly started going through water like a fiend, and consequently peeing all over the place. I filled her water bottle around twelve hours ago, and when I checked on her about half an hour ago it was dry as a bone. It's a 32-ounce bottle! It usually takes her at least two full days to drink that much. She's also been filling her litter box much, much faster than usual, and has been having accidents. She hasn't peed outside her cage yet, but she's been peeing on the floor of her cage quite a lot, which she hasn't done in ages.

Any ideas? I plan on calling my wonderful vet on Monday, but I thought I'd gather opinions here as well.