This is a classic loose neutral scenario, in my opinion. Check in your service panel and tighten all the neutral connections on the neutral bar. Also tighten the service neutral lug, if you can. I suspect that the circuit that you're having trouble with is a multiwire circuit (ie., leaves the panel in a cable that contains a black, red, white, and ground. With the black and red being on two different breakers), which may have a loose neutral. The other end of this multi wire cable may have a loose neutral wire nut also, where the cable splits out into it's two respective circuits. Use the "search" tool above and search for the term "loose neutral". There have been several threads on this in the past which will prove very useful to you.
By the way, you should not have 120 between neutral and ground. Often, there's a few volts, but only a few. A great voltage (up to 120) pretty much identifies the loose neutral problem.
"Nothing is foolproof to a sufficiently talented fool"