I traced the problem back to the -15 V regulator. There appears to be 25 ohms between the output terminal and ground.
I checked each of the capacitors between these two nets (i.e. I removed them and then measured their capacitances) and everything checked out.
Finally, I removed all components between these nets. The 25 ohms remained. I did other things like removing nearby diodes and also the 9-pin D-sub, all to no avail.
I checked one of unstuffed PCBs and it did not have the same issue. I started stuffing it (just the power management), and it was able to regulate to +/-15 V with no problem. This suggests that there is something wrong with the first board internally.
Tomorrow, I will finish stuffing the second board with fresh parts. I will check for shorts along the way to see if I can catch the demon in the act.
I finished stuffing the first RFPD board today, so that we can begin with the testing. Unfortunately, it seems to have failed its first test, as something shorted when I powered it up. Rich and I traced it back to one of the 15 V regulators, and we are going to use that as a starting place tomorrow.
There was nothing obviously wrong with the board as checked by measuring resistances between terminals that should not be shorted (and vice versa), so we hope that there was just a bad regulator.
Besides that, the board looks sweet!