I checked the diodes and regulators and everything looks good there.
I ran some tests on current draw and found some curious things. Without the nano installed, the pcb draws about 25 mA at 25 VDC. The regulators are cool to the touch. With the nano installed, running my sketch, it draws 135 mA and the regulators get quite warm. Since the Arduino IDE was giving a time out error I thought perhaps there was some problem with one of the reset pins. I pulled the nano and was able to recreate the error by jumping the reset pin to ground and trying to load the sketch. But when it's in the board I see 5 VDC on the reset pins so not sure what to make of that.
I installed blink for a reference and I can see the current switching between 55 mA and 115 mA every second on the DMM. Maybe my meter is too slow to track it, but it seems odd that the on-board LED would pull that much current.
Some other possible sources of problems... Pin 10 on the nano socket is grounded somehow. Coincidentally, Pin 10 goes to the chip select pin of the dpot, and it is active low so perhaps not a problem in the end. I tried loading my sketch with the nano off the board and pin 10 jumped to ground and everything worked fine.
At this point, aside from the excessive current draw the main symptom is that the nano is somehow getting stuck during the load process and won't accept data. I did test pins 1 and 2 and didn't find anything obvious there. It make sense to think there is some connection to the mystery current drain and the nano not loading. I've cut the traces of the 5VDC rail to the reset pins but that didn't solve the problem either.
Sorry if the pics aren't that good...just my phone camera. If there's a spot that looks suspicious I can try a zoom pic to get a better look.