I got my pcb some time ago and tried soldering it. I seemed to manage most of it except for the connector. It's hard to even try to solder the connector without touching and melting the plastic, or maybe I need a smaller soldering iron tip. If I order more boards, I'll be making the pads longer.
So while ruining four of my five connectors, I decided to solder everything else, if only for practice. Finally I powered it up with 2 AA batteries - not 3.3V but hopefully close enough. And got a thin column of smoke from somewhere. Turning it off and on gave the same result again, and the smoke seemed to come from somewhere around the LT1945.
After giving it a rest for a few weeks and looking at the board visually, I noticed that some of the LT1945's pins were shorted (VPOS_CTRL and GND, but I didn't check at that point), so I fixed that and tried again. Still smoke. One of the inductors wasn't soldered properly, so I fixed that, and cleaned or melted some brown stuff (flux?) next to a nearby component. No more smoke after that. Somewhere along the way, I noticed the batteries weren't giving 3V anymore, so I replaced them and finally got out the DC/DC power converter I bought from Seeedstudio a while back and got a much nicer 3.28V.
But the output voltages are all off - I'm getting +0.66V or +0.2V instead of +22/-20V. So maybe the LT1945 really is broken.
I don't really know how to continue debugging from here.
Anyway I ordered more spare parts so I can try doing it all over on a new board.
edit: I forgot but some of the LT1945's pins on the other side also looked shorted at one point, so of course I fixed that. Could maybe better explain what went wrong.