Can brakes immediately stop working altogether due to overheating? And if so, would letting the brakes cool make them work again?

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yes and no , only if they unlock themselves when they are cooled enough with luck .best way is to replace them , they have worn down lo much . Take to a mechanic .

 The E brake (which is a parking brake) activates the rear brakes (FWD,RWD,4WD,AWD) and that is not the same as your brake pedal (which first activates the front brakes for you to stop)  It hardly ever relies on the rear brakes.

 So if your brakes do not work, check the fronts.  You will wear out the front brakes 2 or 3 times before you need to replace the rears ONCE.  Obviously if you could drive then the rear brakes were not set up nor was the hand brake fully applied. If it is fully applied you would still be sitting there burning the rubber off of your front tires and not moving.

.  Rear brakes need adjusting.  So find an area where you can drive backwards 100 feet and apply the foot brake.   Do that several times as the adjusters take some time to adjust the rear brakes as it goes 1 cog at a time. After a half dozer reverse rips your rears should be adjusted up.   There is a screw under the handle of your E brake that shortens the cable and you may have to adjust that so that you get maybe 3 clicks before the rear brakes are on HARD...and you don't move.  Then you set up.  The front are self adjusting so are always set up as tight as they can go and still allow you to roll.

.  You can't heat them up that hot in a normal car.

Big trucks driving in the mountains with a load, can get the drums red hot when it is a long hill down. They stand the most chance of overheated brakes(which is why asbestos is used in the pads because it does not burn and is softer than steel) yet even they do not experience total brake failure.  Their brakes return to normal once they cool off.

.  You freak out about something YOU can't do.  It is not an issue.

30+years mechanic on commercial trucks

BRAKE FLUID is an ESTER COMPOUND that SEEKS MOISTURE from teh AIR and eventually turns DARK BROWna nd loses its proteftive BOILING POINT< adn THEN becomes CORROSIVE and SHRIB VELS internal SEALS and ALSO turns into LIQUID SANDPAPER and GRINDS calilpers down and RUINS BRAKE HOSES and BRAKE LINES! as well! The SMART EUROPEAN car makers INSIST upon FLUSHING and CHANGING the BRAKE FLUID every 3-4 years of use ! If YOURS is DARK BROWN now, you WIL need new parts VERY VERY SOON! GFM CARS are teh FIRST to get ROTT$ED brake lines and HOSES! IN THEORY< by changing ALL the fluids every 80,000 miels or so wil keep nearly ANY CAR running to 500,000 miels PLUS! SO FAR< tho, many JEEP and CHRYSLER e 3.7 and 4.7 engines are made like TINKER TOYS and same for many GM engines as well! ouc an TRUST TOYOTA and AUDI and BMW and BENZ the mos and then there is FORD! FOrd NOW uses 100% EUROEpAN technology and BORROWS suppliers that AUDI and MAZDA use! The EDGE is half FORD and HALF AUDI using ZF trannies and ZF steerign and ZF cooling systems along with AC parts! These are the BEST! >>>ANYWAYS it DEPENDS upon WHO supplied your BRAKE PARTS< and YES once they COOL DOWN it SHOULD go back to NORMAL! GO to a DIY CAR WASh and washt eh BRAKES DOWN as there wil be w buikldup of BRAKE DUST and ThAT is NOT GOOD! it can EASILY be WASHED OUT! I prefer CENTRIC BRAKERS and a few other brake suppliers liek ATE GIRLING> GM cars had the WORST p brakes for DECADES i am SORRy to say! OVER BOOSTED and the X cars used ALUMINUM BRAKE DRUMS at first! ONCE AGAIN Gm did NOT fully test E ANYTHING! They let the GM REPAIR SHOPS deal with thei FLAWS! EVEN NOTIVE that there are nearly NO COLLECTABLE American CARS< except for the last DECADE or so! THere are no collectible X, A, or J cars and mos CHRYSLER CARS WERE best off being f driven DIRECTLy tot he JUNK PILES! And gthen here are the FABULOUS GERMAN and EUROPEAN CARS with OIGNAL ABS brakes and QuATTRO and all the OTHER fetures that SAVE LIVES and get the BEST mielge and SAFETY ratings usually! SAFETY CELL engieneering! if you EVEN los anyone you cared for in a CAR CRASH< you woudl be VERY SENSITIVE to CAR BUILD QUALITY! Many car makers use a BAD design and SLAP ON AIR BAGS and LARGER belt systems! >>>i always O+ WODNERED WHY they do NOT use REVERSE MAGNETISM for BRAKING that has NO FRICTION wearing pargts at ALL! JUST an IDEA! 

The car shouldn't be able to move with the parking brake ("emergency brake" is a misnomer) engaged. If it's a rear wheel drive car, and the engine is powerful enough, it may be able to overpower the fully-engaged parking brake, but it will struggle to drive around. The fact that you were able to drive around with the brake engaged means one of three things:

1) The parking brake wasn't fully engaged

2) Something is wrong with the parking brake

3) The rear brakes aren't working properly, and need service

You need to get your brakes inspected by a certified mechanic immediately. At the very least, your rear brakes are damaged, or they were not working properly to begin with. I can't say whether or not they will work again after they cool down. You need to have your car towed to a shop. Don't risk driving with bad brakes.

It baffles me how someone can be so oblivious, that they can't tell that their vehicle is struggling to move. 

They will sort of work, but you need to take it back to the shop. Yep, you messed up, chalk it up to experience get it fixed and move on, only  more carefully next time.

When excess heat boils brake fluid, the brakes have no pressure to operate until the vapor cools enough to condense back to liquid.

What is an "E brake"? Do you mean the "parking brake? 

Just have them inspected to see how much brake lining material you burned off. Rear disc brakes have tiny parking brake shoes that barely function as a parking brake so they don't last long in the event of a boo-boo. 

1st off...  It's totally normal for new brakes to smell for a little while [usually like burnt toast] when you first replace the pads.

The hand brake is separate from the regular brakes, on most vehicles, and they're always the rear brakes.

The front brakes do most of the work, in normal driving.

Typically, driving with the e-brake on will toast the shoes and can warp the drums, but that shouldn't affect your ability to stop.

If it did, then your front brakes were already gone and the vehicle's been unsafe to drive for longer than you realize.

Either way, if they "didn't work" you'd better bring it in to get looked at.

Not if the pedal is going to the floor.