Sure. You know for example, cats with FIV (Feline AIDS) can live along time. You could not have other cats, but you could have that the one cat for along time.

I rescued a kitten with FIP which is an infectious disease that can be deadly. He lived for along time.

I adopted a three-legged cat named Tripod. He was hysterical and it did not even bother him that he had three-legs.

My current one dog was adopted from a shelter that was going to kill her. She had behavioral problems. She is a great dog now.

BUT....

...I believe if you own a pet, you are required to do everything you can for the health and well-being of the animal. Not getting the pet medical attention, not paying for its medications, etc is abuse.
People should not have pets if they cannot care for and AFFORD them.

That is the same with special-needs animals.