I think Laura is getting it right. Even though intelligence might help people increase their status in a vacuum, highly intelligent people tend to have other traits that might decrease their seeking of status or success at attaining it. It's a tradeoff.
For instance, many highly intelligent people are introverts, yet extraversion is probably an advantage for leadership. There is also a correlation between intelligence and openness to experience. While you probably need decent openness to be a good leader, you will seem wishy-washy if you have too much.
In certain fields that depend on competence, such as science, technology, and some parts of academia (but not others), status depends more an accomplishment and is more tightly linked to intelligence. In other fields, the most intelligent and epistemically-trustworthy people will have trouble ascending to leadership positions.
Of course, within the subset of intelligent people, ascending to higher status might also reduce their epistemic hygiene, as Eliezer observes. (Though these individuals will maintain the same intelligence.)
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: