I've had something like this happen in the past. It should just need a reset, if you haven't tried that already. Plug it into the charging cradle and hold the home button for 7-10 seconds to reset it. Turn the bluetooth connection off from your phone while the Fit is off. When the Fit restarts, you should be able to turn bluetooth back on and reconnect it. If you have tried a full reset, then I'm sorry.. no other ideas. I wish you luck.
A short fix as well.... if the gold/copper contacts on the back of the Fit itself are corroding like mine did, use a stiff pencil eraser and rub at the contacts carefully. If you do this before the corrosion sets in too much, you should be able to prevent it, with regular cleaning.
Thanks for the help, folks. My Gear Fit absolutely refused to charge last night, no matter what I did (different charge cradles, different cords, left it plugged in all night)... but after it was plugged in all night and reset it (15 sec hold), I reconnected the bluetooth on my Note4 and now all is well... I think LOL. I had been trying to figure out a good way to clean the contacts. I never thought about a pencil eraser. I was just wetting it a little and that had helped before but the corrosion has been getting worse. The eraser helped a lot! Thank you so much. I use my Fit for everything and wasn't looking forward to having to buy a new one, as this one's just out of warranty!