That smells like a bad ground side... The light that lights is doing so because it is getting its ground through another bulb (putting it in series with the other filament.) When you try to turn on both lights, you lose the ground path and nothing lights. Troubleshoot that by trial/error or get an elcheapo DVOM from Harbor Freight/Wallyworld/AutoZone. Any ol' cheap one will do for most basic auto electrical troubleshooting (you only need a Fluke 87 for really advanced stuff. Or bragging rights. Or if you like yellow.)
Check resistance (without turning on the light/turn signal/stepping on brake - voltage will fry the resistance section of elcheapo meters - that's why they are so cheap, they're disposable) from the ground side of the bulb to some unpainted car body metal or bolt head. Should read continuity (0 ohms resistance or really close to it). Switch meter to DC voltage, have someone turn on light/signal/brake and check for your nominal 12 volts to the car body from the 'hot' side of the bulb socket pins. If you have 12 V to car body in the right places at the right time and the bulb doesn't light there's a good bet that if you install the bulb (and it's doing its lights/glows dimly/sometimes thing) and then check for voltage on the 'ground' side of the bulb you will see voltage - like close to 12 volts. Then you have to find out where the ground side is open. Or cheat and splice into the ground wire and connect it to the car body somehow...
A side note: Harbor Freight sometimes has elcheapo meters for sale in their weekly/monthly/whateveritis ad flyers for all of $2.75 or so. Of course, they fry easily also because there is no protective circuitry included at that price point (ya gotta pay for that Fluke if you want to abuse it.. ;-) but at that price you can buy a carload of 'em. Sortof.