Unteachable

I had a former student--who is now in graduate school herself, and serving as an instructor--message me today with a fascinating question:

Is there such a thing as a student that is unteachable?

My immediate, almost instinctive reaction was, "Of course not!"

But after thinking about it for a few moments...I wonder.

Here's my tentative, pretty-sure-for-now-but-open-to-revision response: