NO! Both lights will stay on for miles or till the engine is shut off long enough to cool down. It is intermittent but seems to happen most often when the engine does not start smoothly.
It should go on when the brake is on and off when you release the brake.
you need a new brake pressure differential switch
The brake light switch is broken or out of adjustment.
Possibly defective or misadjusted brake light switch.
brake light switch is bad
Try checking the brake fluid. that's what happened to me as well. the light on the dash wouldn't go off. all I did was add brake fluid then took the e-brake off and off went the light. I have a 90' accord but I think 90-93 is basically the same.
Might be a stuck brake light switch its located at the top of the brake pedal
Brake light switch went bad.
In most new cars the parking brake light is also activated when the linings of your brakes wear out.
The brake light switch may be out of adjustment or damaged.
altenator
I have a 1800 vectra and the break light comes on when traveling then goes off, is this the brake shoes or pads wearing out.