Well, not altogether.
The case of “Tan”, the guy who could only say “Tan” (already mentioned by thelurkinghorror) is pretty famous in the history of neurology. His was a pretty severe case, and the first scientifically described case, of Brocca’s aphasia, a relatively common and well studied neurological syndrome. Most Brocca’s aphasia sufferers are not quite this bad, I think, but, in general, they are very limited in what they can say - they do not seem to be able to find the words they want very often - although they may not seem to be too badly cognitively impaired otherwise. (And sometimes they can sing words that they can’t speak normally.)
As with Hodor, Tan wasn’t his real name, but became his nickname because it was all he could say.