LM2776 Inverter is not producing -3.3v with 3.3v input
The breadboard prototype of this circuit did work a couple of times, (maybe 3 out of 50 tries), but those may have just been flukes that mislead me into thinking that something must have changed to make it not work anymore. 

When I assembled the circuit on a PCB with all surface-mount capacitors that adhere to the layout recommendations given in the LM2776 datasheet, it works 100% of the time.

I suspect uin28_t's comment was on the right track - the through-hole capacitor leads and DIP adaptor for the IC introduced enough impedance for the LM2776 to be unable to maintain the correct voltage, and I just got lucky the rare times it did work.