In one bedroom, we set up a relatively small air conditioner and a fan in the same (double) outlet, and a nightlight across the room in another outlet. Everything ran fine from 6pm until 2:30am when these three things went out. This room, we think, is on the same circuit as our dining room, but only this bedroom was affected, nothing else went out. The circuits all seem fine. What did we do and how could we fix it?

If all breakers are ON and Puttng out voltage, and if its on, and that room is on same circuit as Dining, You will need to test Last working, to first broke.
Can be in box, Back to Back.

What Strat is referring to is there is most likely a loose connection in the circuit, probably at one of the outlet boxes, and usually at the last working outlet on the circuit.

The AC load is a large current load that is constant for long periods, and loose connections heat up and loosen worse, may even damage the connection. So look for loose screw connections on a receptacle or a loose splice. This bad connection can be either in the neutral (white) wire or the live (black).

And the answer is... turned out it was a spliced wire somewhere in the circuit. It took twenty minutes for Mr. Electrician to find, and considered himself lucky it didn't take him five hours, he said.