Is it possible for someone with mental illness to break off a relationship with you, then want you back later?

First of all you need to ask yourself if this person makes you happy. Second you need to remember that this will continue through out your entire relationship. It is extremely difficult to be in this kind of relationship because of all the ups and downs. The person with the illness isn't sure what they want and will even talk themselves out of what they want or what is good for them. If you are serious about this person suggest going to a counselor together, maybe between the two of you you can find the happiness and calm you are both seeking. These relationships can happen. It takes a lot of patience, love and understanding. When you feel that things aren't being productive or they seem to continue to spin, its time to step back and rethink. Good Luck!