There are several legal issues with making wildlife into a pet, these laws vary by location and any response on that topic would be out of scope for this site.

It happens that Saturday; I met a woman who's children found a baby cotton tail rabbit and kept it in captivity for several years (I believe she said 13).  It was particularly interesting to note, that the rabbit did not appear to become the friendly creature we expect with domesticated bunnies.  But this is not in itself conclusive, as any animal who is not suitable handled and trained, will not become a "suitable pet".  

I spend a lot of time doing domestic bunny eduction, and interacting with the public who have used varied methods of interaction with a "pet bunny".  In this case I got the impression that the care provided to the wild bunny was neither significantly better or worse then average domestic care, in the end I left the conversation feeling that the experience was not as rewarding for the family or the bunny as one would expect with a domestic bunny.

Again depending on local laws your local shelter may or may not be able to intake wildlife.  You may wish to consider searching for local "wild animal rescue" organizations, and discussing the situation with them.

You should keep the wild bunny separated from your pet bunnies, as most if not all of the illnesses and parasites that infect wild rabbits can cross to your domestic bunny.

In the end if you want a bunny to be a good pet, obviously the best results will be with one of the [domestic breeds](https://en.wikipedia.org/wiki/List_of_rabbit_breeds). 

**Expand**

Last evening I attended a presentation by Jill Argall who is the Director of the [Wildlife Center for the Pittsburgh Animal Rescue League (ARL)](http://www.animalrescue.org/wildlife-rehabilitation-center/i-found-an-injured-wild-animal/) they intake both domestic and wild cotton tail rabbits (*the public often does not know the difference*).  She said that the difference in response to humans was significant between the wild and domestic species of rabbits.  They have good results with releasing young weened cotton tails rabbits as well adults who are brought injured. 

The ARL has both wildlife and domestic animal programs.  Domestic rabbits would transfer to the shelter system, to be placed for adoption.  Wild rabbits would stay in the wildlife system to prepared for release.