Yes, breakers do go bad and so do GFI units. Start tracing the circuit. I'd go for the breaker first then work down the line. Just because you hear the click when you set the GFI doesn't mean it's functioning right. Just a word of caution when measuring voltages in places like these, wear goggles or full face protection. On the hot side of breakers you only have the main breaker for protection and one slip can blow a screwdriver or test probe into bits of molten metal. They haven't figured how to do eye transplants yet.
Al