The biggest problem that I come across with overloaded neutrals results from messing up the phases.
If the original installer used a shared neutral it will only carry the maximum load from a single circuit; but if the neutral serves more than 1 phase the neutral can be tasked with carrying twice (or even triple) of the current on a single circuit.
In a home 120/240V:
Shared neutral circuit connected to Black and Red incoming lines.
During renovation the Red ends up being fed from the Black incoming line.
Shared neutral is now being expected to carry twice as much current as intended.
Neither of the hot conductors is overloaded so the circuit breakers never trip.
Neutral carrying twice the rated current overheats, causes fire, house burns down, bad day for everyone involved.
BTW - This has happened enough times (including fatalities) that there is a movement in the NEC to prohibit shared neutrals entirely.
