Do low voltage or high area invalidate that resistance is inversely proportional to cross sectional area? I have learned from high school and college that, for a conductor of uniform cross section, its DC resistance is inversely proportional to the cross-sectional area. I have no doubt that this holds true in practice, for the purpose of everyday electrical appliances and electronics.
But is there a certain scale of things at which this relationship breaks down? For example, if I put $10^{-20}$ Volts across, does it make any difference whether the cross section is $1\,\text{km}^2$ or $100\,\text{km}^2$?
 A: *

*With a low voltage applied, there is no change to the model.

*With a large area, there is no change to the model, provided the voltage on the two terminals of the resistor is applied uniformly across the entire area. 
If the cross sectional area is large, and the voltage is applied only to a small part of that area on one end of the resistor, then spreading resistance will increase the net resistance of the resistor. This effect will be larger if the resistivity of the material is higher.

*On the other hand, very large applied voltages (and the resulting large currents) will likely lead to nonlinear behavior in real materials due to self-heating of the material, so the uniform resistor model ($R=\frac{\rho l}{A}$) will no longer be accurate.

*Very small cross sectional areas could also result in nonlinear behavior, even with modest current, due to the high current density. And atomic-scale cross sectional areas can lead to quantum mechanical effects changing the current-voltage relationship dramatically.
