NXT Bluetooth "Failed!" when turning on
The hidden button under the NXT battery seems to be a reset button, which has to be pressed for 7 seconds to have any effects. I assume it resets the brick in its factory condition, but I didn't find any specific documentation from LEGO on it.

However, there was an issue a few years back which caused the brick to freeze and click permanently (which was called the "NXT Click of Death"); and the solution included a hard reset using that button. (It's just one out of many pages on the subject, by the way)

Will this solve your Bluetooth issue? I don't know. If there's a problem with your firmware, then it's possible, since you'll need to reflash one anyway (since it will be reset to factory settings). Maybe trying alternate firmwares might shed some light on your brick's condition, too.

If that doesn't solve it, I would advise to contact the mindstorms customer support - worst case scenario, they establish your NXT brick is indeed defective, and they'll probably provide you a replacement.