If I understand the question right, the answer is basically no. Ohms law must hold at any point in the system. Say your load is a 480 ohm resistor. At 480V, it normally draws 1 Amp. If the utility can only supply 1/2 amp (say, because it is overloaded) then the voltage across the load will have to drop to 240V. There is no way the current can be anything other than 1 amp if the voltage is 480V and the load is 1 ohm.
Motor loads and capacitive loads and nonlinear sources and loads make the issue a little more complicated. And if we consider standing waves, then the voltage and current can be different at different points along a transmission line. But I think it would be a very rare case if you measured normal voltage but the utility couldn't provide the current draw.
Edit: I am also assuming you are measuring the voltage with the load connected. If you measure the voltage without the load connected, the answer would be a definate yes.
Steve