When in the throes of depression, with no previous experience with “can’t vs. won’t,” I don’t think one can see the issue clearly due to that “Catch 22″ you often refer to: the depression itself blinds us to the possibility that we have any agency at all in our illness.

However, once supported in an intensive therapeutic environment and exposed to the possibility of self-agency, I think willingness becomes seen as a real option where it wasn’t seen before. And once a patient actually steps out on a limb and tries to apply agency along with the will to get better, and it works, a cycle of positive reinfocement can be created.
So I think there are several factors that need to come together to make the “can’t” into a “can,” and the “won’t” into a “will”:
1) a strong therapeutic alliance; 2) environmental support (e.g. milieu therapy); 3) exposure to the reality that one can truly affect one’s illness in a positive manner; 4) and taking the step to actually create that reality in one’s own experience (which involves another level of self-agency and willingness somewhat higher than the initial willingness to simply imagine/hope for the possibility).

Bottom line, in my experience: it takes an extraordinary amount of knowledge, effort, and willingness on the parts of both the therapist and the patient, coupled with an extremely supportive environment, to bring about any true measure of healing in a person with long-term, chronic/acute depression. But it IS possible.