I had to cancel my reservation at Hyatt Kyoto a few weeks ago due to circumstances beyond my control and the hotel was very gracious and refunded my BRG rate. Perhaps they partially did it because the rate was ridiculously low due to BRG and they almost surely would be able to make way more if another guest booked. And if another guest did not book, the amount they were losing was so negligible.

As others have said however, it's up to the individual hotel.