I just experienced the same issue. My Gear Fit just disconnected and went black. Holding the button down would make the device vibrate, but the screen remained black. Just charged the device yesterday and should have over 50% battery life left. These devices are, in my opinion, too new to be experiencing battery failure.
I put mine on the charging cradle for grins, but still no response. Interestingly, and this may be a completely random coincidence, I checked my phone (Galaxy S4 Actove) for a software update and my Gear Fit came back online. Lost my step count for the day, but my exercise history was still on the device. I can't for the life of me think of a reason why or how the phone could keep the device from turning on, so again...this was likely a coincidence.