We could have an entire discussion on this topic alone.

What you describe is a little unusual in a DC circuit though because even if voltage were to go to zero, the current induced by an inductor releasing its energy will result in a change in voltage since resistance is most likely going to remain constant. Since inductors resist change in current, the current created by the inductor will actually cause voltage as a result of the resistance in the circuit. Also zero resistance doesn't mean infinite current, it means zero resistance, as in zero energy lost as current travels through it. You're still limited by the number of electrons available to be moved (which is where Dave's usage of coulombs, which is the actual measurement of electric charge (not electric potential, volts,) comes into play.) Usually the case of current and no voltage is in AC circuits where inductive loads cause the current to go out of phase from the provided AC voltage and this is referred to as reactive power. This can cause a current draw when the AC circuit voltage is zero but, it can also be the case that a non-zero voltage can have zero current.
I just felt a little more information on the topic was in order as there seemed to be a bit of confusion. Not to say that what anyone said was particularly incorrect, just incomplete.
With that said, I would recommend playing with CircuitLab if it was still free. I'm pretty pissed off that it isn't anymore. It was a fun tool for prototyping simple circuits.
I apologize ahead of time if anyone considers this post off topic.