I just got another batch of boards from OSH Park, and they're not good.  Some of the holes appear to have a bit of soldermask inside of them, and others are drilled too small or plated too thick so the tiny pins of the USB connectors can't fit.  I'm disappointed, but not surprised—these boards are a little beyond OSH Park's design rules, so I can't fault them for not being made perfectly.

To avoid problems like these in the future, I suppose the only options now are to either look for a different fab or stop selling the PD Buddy Wye.  I'll start looking for a fab that can handle such tiny features at a reasonable price.