I work with a lot of students who have learning disabilities; I know it's not mental illness, but I want to make a parallel here. Some students (and their parents) acknowlege the LD, and want accommodation from the school, and often make excuses for the child's behaviour and academic progress. Other students (with support from their families) although they work with accommodations, realize that (or perhaps accept), even though it's not "fair" they need to work harder than the typical students in order to succeed in school.
Still, these students may not be at the top of their class, but they are working "at the top of themselves." And as their teacher, admire them. I know that in life, they will be ok, because they are learning skills and strategies to be successful.
I think it can be the same with any illness, be it mental or physical. With anxiety or depression, I have no doubt that it is more challenging to function day-by day than the typical person. It is not fair, but those who suffer will have to work harder (not unlike those people who struggle with addictions, or those more prone to addictions than others). On those days that they succeed, they should be celebrated, and on days they don't, much grace should be given.
I do believe though, that Christians who are conscience of their sin are held accountable. I am not sure what Paul's 'thorne' was, but he said the Lord told him that His Grace is sufficient.