You have a loose connection. It could be in one of the working or non working devices on the circuit.
A common cause of this problem is use of the back stab connections on receptacles. Put the wires under the screws if the back stabs are used.

"I checked the power with a digital meter"

This is a common problem with digital volt meters.

The input impedance is so high (many tens of mega-ohms) they can read voltage through insulation from one wire to another.

It is often referred to as 'phantom voltage.'

If you plugged a light into one side of the receptacle, turned it on, and then measured the voltage you would probably see a nice fate zero volts.

The bulbs resistance is low enough to make any slight leakage through coupling disappear.

There is a loose connection in the circuit somewhere.

Assuming that your main panel has the proper cover in place, check by feeling of the temperature of various places on the front of the panel. Any place significantly warmer than the other parts should be suspected as being near the loose connection. Loose connections deteriorate quickly. Repair cost escalate rapidly a deterioration continues. Get this properly repaired now.

Thanks for all the replies and advice! I'll check the outlets with something plugged into them next time with the digital meter. I'll also check for a loose connection on the preceding outlets and see if I can find any loose or bare connections.

My apartment has one of those 4+range boxes, as do many apartment buildings in my town.

My CURRENT apartment has been rewired (yes...) and makes sense. However, in the last apartment, there were four "circuits", all of which contained fuses. Three had wires connected, but only two did anything I could ever identify. One went to a single outlet in the corner of the livingroom, the other went to EVERYTHING ELSE.

Where I'm going with this, is that a house wired that long ago to have one of those boxes might not make much sense. There's only four circuits to begin with - two per leg, so if one of them ever got discontinued in the past, there would be one remaining circuit that would be the only load on one side...

And it's not unfathomable that lpaz's livingroom outlets could be that circuit...

Sooooooo there's nothing saying those livingroom outlets couldn't be the only thing on one leg, meaning the range of places where a bad connection could exist extends all the way back to the transformer... and if the wind is really affecting it...

If the problem is with one of the poles of the single-phase power supply prior to the main disconnect, none of the 240 volt loads would operate properly. Such has not been reported.

Buy a cheap analog meter and use that. Digital meters are inappropriate tools for casual users. They can easily mislead someone who doesn't understand their limitations.

I use two 15 watt incandescent lamps in series, installed in pigtail sockets. Yes, I do have a BK Precision clamp-on meter, a Wigginton and other testers. But the two lamps tester often is all that is needed.

Creek, I may have missed something, but it's been years since I've seen cheap analog meters in the mass market stores. Even in China, they're not economical to produce compared with DMMs. Digital meters are ridiculously cheap to make using crummy, short-lived chip-on-board construction. That's why you can buy 'em for 3 bucks or so on sale at Harbor Fright.

I agree with you and Bus though. A beginner is not well served by a DMM. He should own two testers - a small neon bulb tester, and a rubber pigtail socket with a 15 or 25 watt incandescent lamp. Tin the pigtail ends to make probes, or if you're feeling wealthy, buy a couple of test probes at an eletronics store.

A similar and somewhat sturdier test light can be made by cutting the female end off an old extension cord, plugging in a nightlight, and screwing in a 4w or 7w candelabra bulb. I used to recommend getting a length of zip cord and an insulation-piercing candelabra base festoon socket, but those sockets don't seem to be a common harware store item any more. Maybe you could cut up an old C-7 Chrstmas tree string from a thrift store or rummage sale.