Yale knows about outside of Yale how? You implying they suck or something? 
In matters academic, those from the top of the top tier know just about everything . . . and everyone. (Note: Not that they're right.) Thus the comparison to Cooley, etc., while (presumably) tongue-in-cheek is a bit off the mark.
The competition at the tippy-top is between Harvard and Yale. (Many put Stanford in the same club, and one could argue for a half dozen others.) Harvard is nearly three times as large as Yale, and while Harvard dominates in just about every way, those in the know know that it is Yale with the edge, academically. This isn't because of superiority of faculty, library, lighting, etc., but because of selectivity. In that word lies the answer to the question. Whether we
like the question . . . that's another question entirely. = : )
PS: Judges are not selected by quantity. If they were, it's Texas and Georgetown that would be the ones to beat.