Many school districts consider family preferences in allocating students to schools. In theory, this approach provides traditionally disadvantaged families greater access to high-quality schools by weakening the link between residential location and school assignment. We leverage data on the school choices made by over 233,000 New York City families to examine the extent to which the city's school choice system fulfills this promise. We find that over-subscribed and high-quality schools enroll smaller proportions of students from traditionally disadvantaged families. We explore three mechanisms to explain this inequitable distribution: application timing, neighborhood stratification, and the architecture of the choice process itself. We find that all three mechanisms have a disequalizing influence and propose several policy shifts to address this inequality.