Wgander,
Yep, as has been suggested, it's most likely a bad ground on that side of the vehicle. On most domestic cars/trucks, the ground is the socket housing that touches the outside of each bulb. And, that ground can be connected to the main chassis or frame via a wire or, simply direct connections of metal to metal. A good test would be to piggy back a wire along side the brake light bulb as you insert the bulb back into the socket. But, make sure the tip of the wire does not touch the contact points on the bottom of the bulb/socket.
Then, connect that other end of the wire to a good ground. Then, try your lights and see what kind of results you have. If your lights now work correctly, then it's a matter of correcting the ground problem with that particular socket.
Scott
Scott and Karla SDFD RETIRED2004 Itasca Horizon, 36GD Slate Blue 330 CAT
2011 GMC Sierra 1500 Ext Cab 4x4 Toad
2008 Caliente Red LVL II GL 1800 Goldwing KI60ND