When I accept the update it gets downloaded. After that, Android Studio closes without any message. Restarting only works on the 2nd attempt but then it is still 2.1.2 and the update is offered again.
(This on Ubuntu Linux 16.04 in case that matters)
At least in my case this occurred because I had bound the prior executable to the launcher. The other motivation for this untethering of Android Studio from the launcher is why it was failing was never viewable. One I did this and manually started Android Studio the upgrade took place correctly and I am now on 2.1.3.
Where this is important is on Monday (8/22/16) or whenever the Android N Studio version is available I would strongly recommend upgrading via command line verse a launch pad linkage.