I have a similar problem, definitely introduced from the iOS 10 update. My Bluetooth headphones no longer connect properly to my iphone 6s+. When I say 'properly' what i mean is that they say they are connected when you look in the blue-tooth devices list but the audio is not pumped out to them very easily. Getting them to work is arduous and involves any combination of rebooting the device, disconnecting the headphones from the phone, restarting the bluetooth on the device. After fussing like that they will eventually connect. They will also not hold the connection, so they can drop once a connection is established.
I even have problems with my Apple keyboard, not as bad as my headphones, but something is definitely up there too.