That’s going to vary a lot from landlord to landlord. I owned the house I lived in all during vet school so it wasn’t a concern for me. My roommate and I each had a dog. I’d say that landlords who often rent to vet students know that vet students usually come with animals or get them during school and private landlords may be willing to bend rules for vet students and allow more pets. It can’t hurt to ask how flexible the policy is…the worst they can say is no and you keep looking. I’d say apartments are less flexible in general.
But with that said, I do think in general it’s smart not to collect a ton of animals during vet school, especially if you might specialize or be moving around a lot. Obviously if you have three cats now that ship has sailed, but (for people who don’t have any or many pets right now) my advice as someone who has moved around a lot for veterinary educational purposes would be to try to limit yourself to just 1 or 2 personal animals, stay away from commonly “banned breeds” of dogs like pitties, shepherds, huskies, etc. or huge breeds, etc. especially if you need to have roommates. Make it easier on yourself to find housing. When you’ve got seven cats, three pitbull mixes and a Great Dane you’re going to have a much harder time finding housing than someone with one cat or one small to medium breed dog. Your social circle in vet school is mostly vet students so your roommate pool is mostly vet students who also usually have pets themselves. Pet limits can definitely limit who you can room with so be smart.