I've had things appear to duplicate, but only had one of them actually debited - like the store runs the card twice, but only one with all the proper information.
I have heard a few horror stories, but we've used our debit cards as our primary payment source for the last 8-10 years, without having any trouble. I check my balance, and deposit/withdrawl status at least 4-5 times a week, and have never seen a double debit occur.
I do know that the duplicate charge has to actually be debited from the account before it can be disputed. I don't know if the dispute would then be with the bank or with the company who applied the debit.