Ok... I had to go back to the manual and clarify a few points myself here.
Armies trace their supplies from the biggest "supply centre" they can draw a line to through friendly controlled provinces. Supplies are generated by supply depots. So say Prussia has access to three supply regions; Berlin, Warsaw and Danzig, in that order of size. If the army can trace a direct line to Berlin, it will draw supplies from there. Otherwise, it will resort to Warsaw and then finally to Danzig if those two are captured by the enemy or cut off.
So, if you can cut off an enemy from all supply depots, then even the smallest army will eventually run out of supplies. You can prolong an army's ability to stay out of supply through having enough supply units.
I think it is region by region, and you can't draw supplies from every source available. So if the best supply source is Berlin, you're not using the supplies in Warsaw and Danzig. They're just there in case Berlin is cut off.