Looking for some help with this puzzle. I live in an 1950's house that has had some renovations but overall still with some its original guts. We have a small heater in one room and a use a hairdryer in a bathroom down the hall. Both cause the lights in the room to dim a little when in use but rarely blow the circuit breaker. The other day both outlets that these devices were plugged into stopped working as well as a few overhead lights in the same rooms. The circuit breaker covering these outlets wasn't flipped and there were other lights/outlets under this breaker that still work fine. When testing the outlets - they read anywhere from 50-70 volts. If you plugged a lamp into these the voltage would drop to 0. Tried replacing both outlets and still no luck. The wires in the outlet boxes look fine with no evidence of blistering or erosion. Any suggestions?

This sounds like a neutral problem. Start by checking the connections on the neutral buss of your panel box if all is tight work back through the indivual connections in juntion boxes etc.
Be sure to work safe and turn the power off. If this fails call a professional as loose or bad neutrals can cause damage to electronics because of the voltage fluxuations i.e.. 40 volts on 1 leg and 200v on the other inorder to make up the 240volts in most homes.

There is a loose connection or splice, most likely a splice, somewhere in the circuit ahaed of the affected outlets, and can be a hot or a neutral conductor.

Since there are other lights and outlets that work, the problem is not at the panel, but possibly at the last working outlet on that circuit.

The bad connection is also possibly causing heat along with a voltage drop created by the high resistance of the connection, so do not use the affected outlets until the problem is found.

A little follow-up... Turns out that another outlet on this circuit had gone bad (i.e. was black). Interesting thing though, the outlet worked (both plug-ins) with a houshold lamp, therefor I didn't remove the cover to inspect on my first run-through. Thanks for the responses.