Third brake light on my 2003 dodge neon will not work, but 2 other brake lights do. Tried several bulbs.

SOURCE: Brake lights

Check your emergency flasher switch... if it located on the column it could have been "hit" and is inbetween the on and off location, this will cause the turn signals and break lights to malfunction

SOURCE: 2001 dodge ram 1500 brake lights

I would guess the brake fuse ( this does not use the same fuse as the tail lights/blinkers) or the ground for the brake light bulb.

SOURCE: 2001 dodge neon tail lights inop

looking at your schematics and , you say tail lights don't work , you didn't mention if front parking lights and license plate lights are working , they all come from the same source , a black /yellow wire , a good clue if front work and rear don't , please advise .if fronts work then you have a open circuit somewhere between junction S118 (near breakout for left headlamp) and rear sockets.

SOURCE: I have a 1998 dodge neon, the rear brake/ turn not functioning

Check all the fuses.

SOURCE: o2 Dodge Dakota Brake lights wont work but bulbs r good.

If the bulbs are good I would suspect the Brake light switch.

Brake Light Switch: Testing and Inspection
The brake lamp switch can be tested with an ohmmeter. The ohmmeter is used to check continuity between the pin terminals.
Switch Circuit Identification
- Terminals 1 and 2: brake lamp circuit
- Terminals 3 and 4: RWAL/ABS module and Powertrain Control Module (PCM) circuit
- Terminals 5 and 6: speed control circuit
Switch Continuity Test
NOTE: Disconnect switch harness before testing switch continuity.
With switch plunger extended, attach test leads to pins 1 and 2. Replace switch if meter indicates no continuity.
With switch plunger retracted, attach test leads to pins 3 and 4. Replace switch if meter indicates no continuity.
With switch plunger retracted, attach test leads to pins 5 and 6. Replace switch if meter indicates no continuity.