I'm not sure about frying the circuit breaker box (although I describe one
possibility at the end of this message), but you can certainly fry
individual appliances.
In the example Carl gave, appliance A and appliance B might survive if their
resistances are the same. But suppose appliance A has a resistance that is
only one-ninth of appliance B. When the neutral is in place, appliance A
and appliance B have a potential difference of 120 volts across each of
them. Now cut the neutral. The applicances act like a voltage divider.
The voltage across appliance A drops to one-tenth of 240 volts = 24 volts.
The voltage across appliance B rises to nine-tenths of 240 = 216 volts.
This could actually fry both appliances depending on what the are. The 216
V across a 120 device will certainly wipe out any solid-state devices unless
the appliance has an internal fuse. On the other hand, low voltage on a
device that has motors in it can cause the motor to overheat and burn up.
Most likely the two 120-volt phases are not "balanced" in the house. This
changes depending upon what appliances are turned on at any particular time.
This means the 240 volts will not divide as 120 and 120, but will divide in
some other ratio that adds to 240. Currently-running appliances on either
side (the higher voltage side or the lower voltage side when the neutral is
cut) are in jeopardy.
Something could happen at the breaker box if any of the breakers are GFCI
breakers because those have solid-state components in them and that portion
might fry.
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
1 University Drive
Bluffton, OH 45817
419.358.3270
edmiston@bluffton.edu