It might depend on the phone you have. Not all phones disconnect calls when you close the phone. That is your responsibility to make sure your internet is diconnected, and I have to agree with Verizon Wireless on that. I read on a US Cellular term and condition that you must physically press the end button to end voice and data connections. Or you will continue to be billed for that call. I always make sure I press the end button to make sure my calls are diconnected. I don't blame Verizon Wireless for your problem, and No I don't work for Verizon Wireless, but I've had them for 2 1/2 years. I had better service from Verizon than Radio Shack, who set up my service.
I changed from AT&T to Verizon 2 months ago and I have some stuff to say.
AT&T had horrible coverage. That was the only reason I left. As far as their customer service I have always believed it was second to none.
This month I got $300 bill from Verizon. When I called to ask why my bill was so high they said it was because I was using the internet so much. Well my internet calls are about 1 minute to check stock quotes. They told me that in order to hang up on the internet I had to press end. Closing the clamshell keeps you online. This is unlike a voice call where when you close the shell it's disconnected.
There is nothing in the literature to warn you about this.
I have spoken to several supervisors about the issue and the attitude has been too bad so sad - you should have known. It's been as bad as someone saturday saying they didnt have the power to reverse charges but they could give me 100 minutes for next month (which I refused). Now, today when I spoke to supervisors they offered me 50 minutes. When I told them I had already been offered 100 they said I should have taken it.
While I have been far more satisfied with the Verizon coverage I have to say that with AT&T they would have fixed the problem and asked me if there was anything more they could do.
As a result, I have cancelled the internet on both our phones saving $10 a month - I figure that will recoup my losses by the end of my contract. At that time I will switch carriers.
I cannot tell you how disgusted I am at the attitude of Verizon. They have to be the most greedy company I have dealt with. At least Sprint lies straight to your face.

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic