DVDdoug:
Ohm's law is always true, so sometimes something has to "give". It's a law of nature with man-made units of measure (Volts, Amps, and Ohms).
Ohm's law is not "always true". It doesn't apply to a vacuum (as in thermionic valves), it does not apply to insulators, it does not apply to gasses or plasmas, it does not apply to superconductors (ie at very low temperatures). It does not apply to quantum-tunneling composites. It doesn't apply to semiconductor junctions and it doesn't apply to living tissues / cell membranes.
Its an approximation to the behaviour of solid conductors, uniform semiconductors and ionic liquids when moving charge carriers interact in random thermal manner with other things.
Furthermore Ohm's law only applies on the macroscopic level and breaks down at the nano / quantum scale.