What happens in the case of a symmetric, non-harmonic generating load is that the fundamental currents cancel each other out when added to form the neutral current -- this is a result of the 120 degree offset between the phases. (Think of it as current flowing from hot to another hot and returning that way instead of returning on the neutral.) There are two cases where this does not hold though: asymmetric loading and triplen harmonics.
In the worst possible asymmetric loading case, the circuit's rated ampacity is placed entirely on a single phase. Assuming that the load is non-harmonic-generating, this means that the neutral current equals the load on the circuit, allowing for equal size hot and neutral wires to be used, and the breaker on the hot side to protect the entire circuit's wiring adequately. (Putting the rated ampacity on two phases doesn't change things, either.)
What's worse, though, is when you have harmonic generating loads spread across the phases. While the fundamental and most of the harmonics cancel, odd harmonics that are also multiples of 3 (3, 9, 15, and so on, called "triplen harmonics" in the electrical world) do not cancel out, leading to a situation where the effective neutral current is higher than expected -- it can be even higher than the circuit ampacity as these harmonics are coming from loads on all three phases and returning on the neutral. The resulting high currents can overload the neutral alone, leading to a fire hazard and the need for a doubled or oversized neutral to counter this.
(They can also overheat the common delta-wye type of power distribution transformer, but that's neither here nor there for this problem.)