I think it depends on the nature of the problem in addition to the perceptions of it by the person who suspects she has it, and how she perceives it is perceived by others as well. For instance - compulsive lying - there is a HUGE stigma against lying in our society, and there is a large element of shame that goes along with it. So where someone might suspect that they have a real problem with lying, they may not want to seek help for fear of being stigmatized further.
Others don't realize they have a problem - per their specific disorder. And, as you said, some people do not like to be categorized that way, and then there are others, still, who don't believe some disorders even exist.
For some people, and more for those that you are very close to, I think it's very important that they know you will not treat them/think of them any differently should they decide to get help and an official diagnosis. Some people, like my father, thought that having a disorder was some kind of "ultimate weakness" that needed to be suppressed and blotted out of existence. All it took was a clearing of the air and a very straightforward talk (in his own style) to assure him that it's all right.
There is not enough said for emotional/social support these days, I think. It seems that as "individuality" is valued more by more people, we're then expected to be able to go at and deal with absolutely everything alone and in silence. It doesn't need to be this way.