This is a very touchy subject and this is were the tenant has to do his due diligence.
This example is for here in Ontario and concerns dogs, but some states may have something similar concerning other types of pets.
A signed contract(lease) that states no dogs has absolutely no value in the courts eyes and you can not be evicted if you choose to bring a dog in. Even if you bring the dog into the home months/years after the fact, as long as the dog is not causing any undue hardship.
It's called "PETS AND "NO PETS CLAUSES":
http://www.ontariotenants.ca/law/law.phtml#Q6
Q6: The landlord says I must either move out or get rid of my pet; Do I?
A6: Only if the pet is dangerous, causes allergic reactions or causes problems for other tenants or the landlord, must you get rid of your pet or consider moving elsewhere as per Landlord application to terminate tenancy based on animals.
Even if you signed a lease with a "no pets" clause, if the pet is not a problem for anybody they can not enforce it; such no pets clauses are invalid under the law.
You do not have to move or get rid of the pet unless the Board issues a written order to do so.
14. A provision in a tenancy agreement prohibiting the presence of animals in or about the residential complex is void. 2006, c. 17, s. 14.