Yes, as I understand it, the support is always cut if the supporting location is attacked, unless the attacked originates from the province where support is given to. Supports for supporting units do not prevent the cutting of support.

An attack on a supporting unit, whether the attack succeeds or not, will cut the support (either support hold or support move), HOWEVER you cannot cut a support on an attack against you, IE:
Player 1:
Rum - Bul
Ser S Rum-Bul

Player 2:
Bul-Ser

Even though bul attacked Ser, which was supporting Rum's attack, the support remains, since the support was to an attack ON bul. Make sense?

In your case, even though the attack doesn't displace bulgaria, the support is still cut.

I'll have to double check, but I also think there is a rule where you can't cut your own troops support, but, honestly, if you are thinking about that, then you are just screwing around, anyway ;)

Pepijn is correct. An attack from anywhere except the target of the support move will cut the support.

@fortknox: You can't cut your own troops' support, no. This could be useful if you want to use a supported move to bounce a supported move, though a support hold is probably better in most cases (since it requires one fewer unit).