Depends on where
In the US, IF you have status with the hotel chain, you will probably be allowed to stay, and they will "walk" some poor sucker who booked on Priceline.
In Europe you will usually NOT be allowed to stay the additional night(s), even if you have status with the hotel. A Marriott Rewards Plat at a full service Marriott in London that deals with lots of Americans? Probably they will work it for you. A three-star in Paris where you are unknown? They will probably make a couple of phone calls to try to book you elsewhere, but that's about it.