As much as I like pets myself, I know how cats can go around peeing on things to mark their territory,  and how bad that smells, and how dogs can chew or claw trim, and have accidents too, and what an odor pets create when they aren't well groomed.

But all in all, if you do your due diligence, and a tenant turns out to be bad, I think it's just a bad accident, like a car accident.

Only after the tenant fled the state was it discovered the extent of trapped animal damage (horrific to say the least). 

Who really cares if the Germans think we are unkind to our poor .. these are the same folks that stuffed millions Of men women and children into gas chambers To die horrific deaths  .

People don't get multi-million dollar awards because of no-fault accidents.So, limit your liability by removing problems you could get sued for - make sure the electrical is to code, and there are fire extinguishers, provide outside lighting if it's a bad neighborhood, take care of pest control, make sure the roof won't collapse, that stairs are in good order, that there are no swimming pools kids can get into.I just can't think of any "terrible injury" that could have been caused by you, if you just take care of the property.And, of course, get a reasonable amount of insurance to cover you for any accidents that may happen, that a tenant may file a claim against you for.But, to put things into perspective - I managed 25 units for 8 years as a resident manager.  

And not all settlements come from car accidents.

Avidyne Integra  and it failed 300 feet above a marine layer when I was making the ILS approach into North Bend... first time ever I had stuff go wonky on me at the exact wrong time.. 30 seconds later and it could have been the big red handle pull and a splash landing in the ocean LOL.. and since I was not expecting that approach I did not have my life raft with me... you know the links in the accident chain... good calm advice for our lady and her 11 gates running on her property... as always the voice of reason.. !!!!

Assuming the tenant is reasonably responsible (sweeps up the hay that escapes the cage, cleans up any accidents) they shouldn't pose any problems.

Maybe they were in the trunk when the accident happened.

You will continue to do it and may or may not get caught again before you have a accident.