Hello,
Lights in the hallway became real bright then blew out & a power strip melted.
Wiring in house is about 10-15 years old.
Did some investigation and here is what I found out:

The neutral wire in the subpanel for the lights was burnt and disconnected. The neutral wire is in 14/3 romex (ground,white, red, black).

The red & black wire are connected to circuit breakers opposite phases but not connected to a double pole breaker (breakers did not trip.
Since the neutral burnt and not connected to the neutral bar and the breakers did not trip and were still on, I put my volt meter to a receptacle and I got 240 volts between the hot and ground and my tester showed hot/ground reverse.
Got 240 volts at the switch box where the light bulbs blew.
I reconnected the neutral wire and turned on the first breaker. Checked lights and receptacles and all seems fine.
Turned off the first breaker and turned on the second breaker. Checked lights and receptacles and all seem fine.

I am afraid to turn on both breakers at the same time.

I am looking for the box with the other end of the 14/3 romex that comes from the sub panel.

Has anyone come across this problem or any suggestions?

Thank you in advance

If the 14/3 is wired as you described, then if neutral opens, you now have 240 VAC where 120 VAC should be.

Unless you have been playing with the panel box then for Neutral to burn and separate as you describe scares the devil out of me. Was there no bad weather at the time or something that precipitated the failure?

Personally, I'd be on the phone to a licensed electrician, now! The panel box can also be the kill zone.

Since the 2 circuits associated with that neutral are on opposite phases, Neutral may have burned due to being loose.
If both circuits were on same phase, Neutral could have been loaded to 30 Amps (15X2).
That is not the case. Reconnect white, or pigtail the white to eleminate burnt. Some screws will seem tight on wire, but really are tight threads giving the impression of a tight connection. I would reconnect white, Have TV or Appliances disconnected, Bulbs are cheap.
Then with both breakers back on. Measure voltage at recepticles.

This is exactly what Strat said. The neutral was loose and the resistance created the heat, which burned the wire.
I've seen it many times.

You will not automatically get 240v where you should 120v. The neutral is what stabilizes a multi-wire circuit and gives you an exact voltage line to line and line to neutral.
Without the neutral you get the up and down voltages as soon as a 120v load is introduced.