Recently my original Microsoft Band split and I returned to Microsoft, they send me a band new Band and I was impressed with quick turnaround. When I got my new Microsoft Band I paired it up with the Microsoft Health app and it all look ok until I found after a couple of seconds it would drop the Bluetooth connection to my Lumia 1520 running Windows Phone 8.1 so it never pushed any notifications to the Band despite it looking like the connection was ok.
It turns out there was a simple fix and this works with the original Microsoft Band, the new Microsoft Band 2 and Windows 10 Mobile. In the Device Hub (which is found in Settings) there were two Microsoft Bands listed which I think was confusing the Microsoft Health app (and 3rd party Band apps). I removed the old Microsoft Band from the list by pressing and holding on old Microsoft Band listing in the hub and selecting remove. I then repaired the Microsoft Band with the Health app and it worked fine. I know this worked for me and for others so if you having a problem it is well worth trying.