when does the neutral wire in a 3 phase system carry voltage?
"Carry voltage" technically is a very loose term. 

Voltage is relative, meaning it must be measured relative to another point, such as ground or another conductor.

All conductors have resistance and obeys ohms law. When current flows there will be a voltage drop throughout the length of the conductor. This happens in all conductors disregarding the systems involved.

In well balanced 3 phase system the neutral current will be very minimum compared to the phase currents. That explains why voltage drop is almost zero in that wire. 

Imagine a thin straight pole is poked into earth perpendicular to ground and tip of the pole is attached to 3 ropes and pulled from three equally spaced (120 degree) directions. The pole tip will stay stationary even when pulled with great force. That's what happens to the neutral in well balanced 3 phase system.

But when a rope a slightly slag or exert more force than the other two, the pole tip will move. Thus, imbalance 3 phase system will have neutral current and the proportional voltage drop along the wire.

In the real world, there is no perfectly balanced 3 phase system. That is why most of the time when neutral voltage is measured against the local earth, there will be a voltage, mostly around 6 volts for the 415 volt 3 phase systems.