I can't give you any advice as to whether or not you'd be liable, but is there any way you can discreetly check what bank details are on your account. e.g. can you check it on-line?
Also, does the bill definitely state that the previous month's bill was paid? i.e. does it show the amount due for the last bill and then show that this was paid? Or is the bill simply for the service provided in the billing period?
If it's the latter, then your case would be weaker, but again, I want to point out, I'm not an expert.