Most of the time not. If it's something that's not obvious to them, they won't think about it. Dementia is a good example. 9 out of 10 people will not admit they have it! But some do know and try to do something about it. It usually takes a couple of people to be around a person to notice any changes. Some are obvious and some aren't.
I work with special Olympics and most all if them know they are intellectually handicapped. As a matter of fact, it was an athlete who started the campaign "spread the word to lose the word" referring to the term retarded.