Hey Bayou Bob, welcome!
It's true that if your switch is original, or a very old replacement, it probably still has the small glass fuse clipped to it. But if it's a more modern replacement (like in the last 20 years or so?) it's probably the updated newer version with the internal circuit-breaker.
Sounds like it might be older though, since the breaker is a self-resetting type so once something overloads it and stops working, it should work again at least for a moment when you try again. Unless it's a dead short, or a dead circuit-breaker, in which case you'd probably get zero joy at any time.
Of course, it could just be coincidental failures too. Your brake lights and turn signals are the same wire from the firewall back, and the same bulbs. Did you check to make sure the bulbs were good?
Your brake switch and your turn signal flasher could have failed also. Both components do fail. Not sure how the '66 was laid out as far as the brake light switch is concerned, but most Early Broncos came with a funky strange switch mounted to the top of the pedal arm at the rod pivot. Funky, but the darn things work really good when they work, so no use complaining I suppose.
I can't remember if your '66 would have had a plunger style (like most of us are more familiar with on other vehicles) or the Ford-unique one on the pedal.
Rear tail-light sockets also get old and rusty. Is yours an early model with no backup lamp lenses in the tail light lens? Just red with no clear? Or do you have backup lamps?
Either way, time to take the lenses off and inspect the sockets for rust and bent up or broken contact springs.
Do you happen to have dual exhaust? If it's a V8 lots of people put dual exhaust on them and the driver's side tube or muffler can get close to the wires on the frame. If they get buggered by the heat they can give you the symptoms you have.
Easy to run down the wires and look for crimped, broken, melted or otherwise compromised wiring.
But another common thread here is the turn signal switch. New or old, they fail often. Your brake light power comes out of the brake switch and up through the turn signal switch so it can go out on the same wires as the turn signals and so the turn signal won't mess with the opposite brake light. Letting both do their jobs.
So it's not just one thing that it can be. Especially with a vehicle this old!
Good luck.
Paul