If you suffer from a disability (including a mental disability such as anxiety or depression) you are allowed, under the American Disabilities Act to have an emotional support animal in your housing even if there is a "no pet" policy (these animals do not have to be formally trained, they can be just animals that help you feel better). You have to get a note from a therapist or a physician but if you actually suffer from a disability this should not be a problem.
Also, managers are not allowed to charge an additional pet deposit for the service animal (a "pet" who helps in an emotional or physical way is no longer a pet but a "service animal") just because they think it might cause damage. You are liable for any damages your animal causes, but they are not allowed to assume.
Here is a website with a sample letter and much information about court cases involving this topic.
And as Bob Barker says, spay or neuter your pets. Adopt from the pound or SPCA or rescue organizations if you can. They make wonderful animals and 1/3 of all animals in shelters are purebred animals. Food for thought.
EDIT: But please be aware and respectful of the fact that the service animal community is concerned about this being abused and making things harder for people with serious service animal needs.
Credit: From a post in the ucdavis community.