It depends...
It depends on the size of wire. (smaller wire, more heat)
It depends on the number of coils. (more coils, more heat)
It depends on the amount of current going through the wires. (more current, more heat)
It depends on how long the high current flow lasts (longer time, more heat)
If you have 30 amp service, and run the A/C and the microwave and the tv at the same time for an hour or so, maybe the wires will get warm.
If you have 50 amp service, you can run everything for a longer time.
Personally, I would not worry about it. While it may have happened 50 or 60 years ago, and the story is still going around, I have never heard of it happening recently. If it was truly a hazard, you can bet there would be warnings posted all over the place!
