That’s a very important question – in that it could be devastating to a client to be “fired” by his/her therapist. It’s tantamount to having your parent dis-own you, because of the transference.
After close to 40 years of doing therapy in private practice I think I have told a client I couldn’t keep working with them 3 times: once, the client was refusing to engage with therapy and only wanted to “use me “ much the way he used prostitutes – to feel better emotionally – and refused to have even have an area he wanted to improve himself about and ‘work on’. Another time it was very similar with a woman, and a third time I was seeing a man for the first time and intuitively felt in danger being alone with him in my office. I later heard he had attacked a female nurse in a hospital situation. In the first two examples I considered what I was doing to be therapeutic – in that the impact of having me refuse to continue seeing these people was my *final* BIG statement to them about the importance of dealing with their behavior.
All of which is to say – if you take someone on as a patient and you are not legitimately over your head with them – you have a moral obligation to them to work with them. It’s really OK to admit to a client that you don’t know how to help them and refer them to someone you think can help them. So yes, I think some therapists probably do give up on clients. Hopefully not often. It’s important for therapists to learn how to “size up” clients who show up at their door and decide if they want to work with this person or not.