I think the biggest misconception when it comes to URM admissions is that there is some shortage of applicants. Just because a school such as Stanford actively seeks a diverse student body does not mean that they have a problem attaining that goal. Schools may aim to hit a certain percentage of AA students for example, but they have a rich applicant pool to draw from, so why is everyone so surprised when a high stats AA student does not get admitted to every school they apply to?
Here is some data from the 2016 admissions cycle:
Columbia: 4606 AA applicants, 174 AA enrollees
Cornell: 3928 and 300
University of Chicago: 2659 and 126
Yale: 3171 and 148
Tufts: 1286 and 64
Duke: 3877 and 186