I had this same problem when i went to get my car inspected (not sure how long I was without fully functioning brake lights) but the guy had a real simple fix. Apparently sometimes the grounding wires on the lamp housing become loose, try tightening those to see if that does it.
You either have a short in the power wire, meaning it is disconnected, frayed against other wires or a ground problem. You can test both the power and ground wires with a simple test light. If you "ground" the test light and figure out which bulb is supposed to be the brake light. If you remove the bulb and probe where the bulb seats to each of the contacts and one of them lights up when the brakes are depressed, you have power to the bulb. This means that either you have a bad ground or you have a bad bulb.
For those unfamiliar with electrical terms a ground on an automobile is essentially the body. Anything metal on a vehicle is "grounded." They do this to eliminate an extra set of wires. Since metal conducts electricity, and most car bodies or frames are metal, they connect the negative battery cable to the vehicles frame and all the accessory grounds are connected to it as well.
So if you need to run a new ground, you figure out which wire needs to be a ground and simply scuff a piece of the body and attach a wire using a ring terminal found at any auto parts or electronic store.
Have you heard of KISS!!!
Keep It Simple Stupid!!! I would check the bulb!
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