SOLVED: Other Bluetooth-connected devices are the cause. They may be interfering/using too much bandwidth.
Before going on to other solutions given here, simply try disconnecting your Bluetooth mouse and/or keyboard while music is playing to your headset. The result will be instantaneous!
I had this same problem and tried all the solutions given in this forum without any result (including uninstalling/reinstalling the Bluetooth drivers which is always unsettling).
However, after reinstalling the BT drivers, when reconnecting my devices, I decided to reconnect the headset BEFORE my BT-connected mouse. Lo and behold, the headset sound was crystal clear. Then I reconnected the mouse, and the issue restarted.
Also, one of the solutions give is to disable the Hands-Free profile from the Sound settings. This will have no incidence on my solution; both profiles work just fine. My problem was really just the mouse.
Hope this helped, and if it did, please "like" my solution. I need the votes :)