I think the overjustification effect might be at play.
The overjustification effect occurs when an external incentive such as money or prizes decreases a person's intrinsic motivation to perform a task. According to self-perception theory, people pay more attention to the incentive, and less attention to the enjoyment and satisfaction that they receive from performing the activity. The overall effect is a shift in motivation to extrinsic factors and the undermining of pre-existing intrinsic motivation.
In this case, the reward is status. It's important to note that the person must anticipate the reward, though. People might explicitly seek status, but subconsciously seeking status might provide enough anticipation to create the effect.
I am taking Eliezer's definition of "stupidity" to mean increased incompetence in the field wherein the person gained status. In their field, we would expect high competence. Decreased competence in their field would come about from diminished interest in that field, from the overjustification effect.
Michael Vassar once suggested: "Status makes people effectively stupid, as it makes it harder for them to update their public positions without feeling that they are losing face."
To the extent that status does, in fact, make people stupid, this is a rather important phenomenon for a society like ours in which practically all decisions and beliefs pass through the hands of very-high-status individuals (a high "cognitive Gini coefficient").
Does status actually make people stupid? It's hard to say because I haven't tracked many careers over time. I do have a definite and strong impression, with respect to many high-status individuals, that it would have been a lot easier to have an intelligent conversation with them, if I'd approached them before they made it big. But where does that impression come from, since I haven't actually tracked them over time? (Fundamental question of rationality: What do you think you know and how do you think you know it?) My best guess for why my brain seems to believe this: I know it's possible to have intelligent conversations with smart grad students, and I get the strong impression that high-status people used to be those grad students, but now it's much harder to have intelligent conversations with them than with smart grad students.
Hypotheses:
Did I miss anything important?
Having achieved some small degree of status in certain very limited circles, here's what I do to try to avoid the status-makes-you-stupid effect: