I have a friend who is like this. She’s just stopped returning phone calls before or stopped answering texts or emails or FB messages, for months. And then she’ll start responding again with no explanation. We’ve known each other since preschool age, and I realized a while back that in the entire history of our friendship, it’s been on-again, off-again, and it’s always been her who’s been off and on, not me. When I realized this I choose to simply accept it as is, and occasionally I try to contact her, with no expectation of a response, and if she does respond, then we chat a bit. When I’m in town (she lives in my hometown where my parents still live) I try to get together with her, and usually she doesn’t respond, but occasionally she’s interested and we get together and have a nice chat. I might not be as accepting with someone else, but she had a really rough time for several years with an abusive husband so I give her some grace, and just accept that most of the time she’s going to be “off” and I don’t take it personally. I don’t put much effort into the friendship at all though.
I think in your shoes I would be inclined to say (whether in person, letter, whatever works for you) something along the lines of…while I really enjoy our friendship, this is the third time this has happened and at this point it is really difficult for me to open up to you or feel close to you because, whatever the reason, you have shown a pattern of abruptly dumping me and then wanting to apologize and make up like nothing has happened. Let her know that she has hurt you and it is going to have a negative effect on your feelings towards her as a friend. And then decide if the friendship is valuable enough to you for you to be willing to continue it, but understanding that if you do there’s going to be a barrier between you for a time, probably a long time, because you don’t fully trust her.
With the friendship that I ended recently something happened a few years ago that really hurt me, and it took almost a year for me to be willing to have much contact with her again, and longer to start to trust her again…and still eventually due to various issues I decided the friendship wasn’t valuable enough to continue.