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It's implied in earlier comments, but I would look specifically at the stop light switch, which is the device under the driver's floorboard that closes the circuit to the brake light when the brake pedal is pushed. There may be some kind of high-resistance grounding occurring when the switch is actuated that would cause a current drop in the other lamps. You can buy a new stop light switch for under $10 (the nice ones are $50, but for troubleshooting purposes the cheap one would suffice).