Electric field required to split water into oxygen and hydrogen
In part, there is no abrupt cutoff because the gradient of the field is not spread evenly across the electrolyte: the gradient is steep at the interface between electrodes and the electrolyte, where gas evolution actually takes place, not in the body of the electrolyte. Note where bubbles arise in this video.

However, if the conductivity of the electrolyte is low, whether due to electrode separation or low ionization, then electrolysis proceeds more slowly, perhaps imperceptibly, though there is no complete cutoff point.