Try tightening all the neutrals in the breaker panel then checking all the connections on that circuit. Make sure the receptacles aren't speed wired.

Check the connections at the receptacle that DOESN'T have the problem... look for a loose white wire.

Sounds like aloose connection inthe bedroom. Could be in the good receptacle or one of the bad ones or the light.
Most commonly this is caused by using the backs tab connection. MOve allthe wires to the screw terminals if the back stabs are used.

Being a rented house this should be looked by the landlord's electrician. Legally you are not permitted to work on electric in a rented house.

you should first find all of the loads on the circuit, by turning off breakers 1 by 1. you'll have a better chance of finding the problem if you know which ones are involved. this is pretty likely a loose neutral in a 3 wire circuit, so you'll also want to know which is the other circuit.

i don't consider this a beginners problem, but if you're willing to pay later if you can't figure it out, then go ahead.

rick

It turns out the problem was with a receptacle in another room. There was one common wall with the master bedroom so the receptacle was on the same circuit. The outlet was well hidden behind a desk, but the second I pulled it out of the wall I knew this was it. One side was badly burned. The wires were so brittle IÂm surprised it carried any load at all. It looks like it was burned from the inside out. I would guess they nearly had a house fire. There is no telling what caused it, but I feel better having replaced the breaker.

Were the wires on receptacle screws or push-in backstabs?

Arcing at a poor connection is unlikely to trip a conventional breaker, as there would never be more than rated current. It is why the newer code versions require arc fault breakers in some locations.