I think it depends on what Bank you use here. I found that a majority of them have a problem with HSBC cards and the bank in Bansko confirmed this, however, the bank will allow you to take out money from a cashier from your credit card if you come in with a Passport. But, i found a machine near the bottom of Bansko that allowed you to use your UK debit card in the machine. It was a Greek bank of some kind and worked fine..
It's due to the fact that most ATM's can't cope with cards with a chip in them. They are a bit behind the times in that respect. I don't know about the other resorts but in Borovets, use the ATM in the Flora Hotel. That worked for us. We found it helpful to just go into a bank and ask somebody, that's how we found out what to do.
I have a Nationwide debit card and HSBC card and had no problems in Bansko. Down near the old square there is about 6 banks and the 3 i tried accepted my cards
In Bansko the cash machines are outside Kempinski Arena and Straxhite hotel, both dont work if you have a chip and pin card so you will have to make a trip iinto the old town!!