"Have you ever had an adult student who you knew would never be able to play the piano the way it was meant to be played?"
I don't know what you mean by "meant to be played." I have taught many adult beginners who I knew would (probably) never become a virtuoso given their differing circumstances. However, I have never taught an adult who I believed was not capable of learning to play somewhere between decently and very well.
"Would you ever consider allowing that student to continue even though you knew there was little to no hope?"
I don't consider myself in a position to "allow" any of my students, especially adults, to do anything. No, I would never suggest to any adult student that they should quit because of lack of progress. I have had students begin lessons with the idea that they would be able to start at an intermediate/advanced level and with those students I always provide music at a more realistic level; I don't like adult students having unrealistic expectations and playing above their level. That said, I will work with any adult student until they get it, for as long as it takes to progress.
"Is there a way to tell that a person is not capable of learning to play the piano beyond the "forever beginning" stage?"
No, I don't think there's some definitive way to tell. There are some factors that I think help adult beginners, such as having SOME musical background (even if it's on a different instrument, lessons as a kid, etc), healthy fingers (I've had a few older students whose fingers were so stiff that I knew they would never be able to play certain things), and time to practice (I've had the least success with adults who have small kids or full-time jobs). In my experience, I have not yet seen an adult beginner progress beyond an amateur level, but amateur does NOT imply bad!
+1 Nice post, catpiano. I agree with all that has been said by other posters thus far as well.
From a teacher's perspective I do wonder if some students will no longer be able to step up to the challenge of the next skill or level of difficulty in a piece, and that if at some point I will just have to make lateral moves. By lateral I mean finding music that isn't challenging but maintaining what they already know how to do. Lateral isn't bad, however, and of course students progress even when playing more pieces at a certain level. There's just a part of me that wants them to progress forward and I have to be careful of that. I guess I get ambitious for my students because I want to see them progress, but there are different ways of measuring progress other than difficulty of pieces played.
Even with working within a student's level of playing, however, there are some nuances that I know just won't happen. It seems with some people there's only so much they can handle at one time, and so if they get the notes and rhythms correct with pedaling, that's about all I can ask. I struggle with the idea of even bothering to address things like phrasing and dynamics. So I can really relate to the poster who talked about students like this requiring more energy than others.