I had this same issue with my CRX. With that car (and likely the NSX as well) they drill a hole in the brake pedal and fill it with a plastic piece, presumably to quiet things down. When the plastic part breaks, the brake pedal no longer touches the switch even when the brakes aren't on, leaving the brake lights on all the time. I grabbed a nut and bolt and plugged the hole, problem solved.
Like jwmelvin said, a short to body would cause a fuse to blow. For this to happen you'd need a short that would bypass the brake pedal switch.