I've come across this line of reasoning before (in reference to the study with math problems/telling kids they're smart versus hard working) but I'm not sure if I believe this is particular to being designated as smart. In fact, I would imagine that telling students that they're not in the 'smart' range, or having peers designated as smart while they are not, is more damaging to academic motivation/leads to more fears and emotional distress. I think both are just manifestations of an underlying fear of being judged, regardless of the expectations - fears of not living up to high expectations, or confirming low ones, or in general moving from a comfortable position of uncertainty in one's skill/intelligence (leaving open the possibility of it being unexpectedly high) to a better defined and therefore less appealing position.

(you might argue that labeling a child as 'not smart' or at least not singling out a child as 'smart' might motivate them to prove the evaluation wrong - but I think those cases are by far the exception rather than the rule)

Now, the label of 'hard working' is on a very different scale from intelligence; I can understand how calling kids 'smart' rather than 'hard working' could decrease their performance. Nevertheless, over the course of our academic lives I think we all get a sense of how we fit in, intelligence-wise, even if we aren't explicitly labeled, from grades, peer performance, parent/teacher expectations and evaluations... the smart ones fear being 'found out' as dumb or tarnishing their reputations, the 'dumb' ones fear (even more, I would imagine) the criticism or negative evaluation of their work they expect to be forthcoming, the average ones fear being confirmed as merely average...