Can two circuits share a neutral?
Only if it is a split 220 volt circuit as Steven points out and it It must be on a two pole circuit breaker. This ensures that the circuit is two different legs and not just one, also the entire circuit will be disabled with only one breaker. Each hot leg returns on this shared neutral. The reasoning is that no more than the rated (breaker) current will pass through the neutral in this configuration. The two "Hot" legs will tend to balance the return current. Example: If both legs draw 5A, then no return current passes through the neutral. If leg 1 draws 3A and leg 2 draws 10A then the difference, 7A is returned through the neutral.

It is NOT permitted to share a neutral in any other situation. If you were to share a neutral with two breakers on the same leg of a panel, both circuits could draw the breaker limit (lets say 15A) making the shared neutral as much as 30A return current! That will exceed the limit of the wire size and could cause a fire. Also, GFCI and AFCI breakers will not work on a shared neutral circuit.