Could another signal be sent down the length of the neuron cell if the sodium ions and potassium ions were not returned to their original positions?

The answer is NO. When a nerve is depolarized it cannot transmit an impulse. It has to wait for the never cell to repolarize. This occurs when K + ions and Na + ions are pumped out of the cell by ATPase until the resting potential is achieved. In most nerve cells, the resting membrane potential is usually - 70 mV.