Does Verizon stop my Wi-Fi from working?

I have an iPhone 4 through Verizon. I also have an iPhone 3GS that is no longer connected with a carrier, and a laptop, and my mother has an iPad. For some reason, my iPhone 4 can connect to my Wi-Fi, but will not load anything. The other 3 devices on the other hand will connect and load things just fine. Often I get too frustrated and just shut off the Wi-Fi availability on my iPhone 4, and hope that 3G stays on long enough to use it. In fact, I'm typing this question on my iPhone 3GS.
I have tried every method of resetting my devices: resetting my modem and router for the recommended period of time (even for a few minutes or more), resetting my phone, and even moving the modem and router to a different location in the room, but no luck. I even reset both phones and did a side-by-side comparison to make sure I wasn't just in a "dead zone." I attempted to get help from Verizon, but all they wanted to tell me was "you're in an area not covered by our towers." Surely tower coverage doesn't change from room to room within my own house.
But back to the point, my Verizon iPhone does not function through Wi-Fi, but my other devices will. I know I may be thinking too deeply into this, but I know Verizon offers their own Internet service, so is it possible they are doing something to interfere with my Wi-Fi signal?

Additional info: My Internet provider is SuddenLink, which uses cable Internet. Also, we recently got a home phone again and it also is connected through the same cable. The cable runs from the wall and into a splitter; from there one cable connects to the Internet modem, and the other connects to the home phone modem.

Verizon doesn't control your Wi-Fi radios in the phone. They only provide the service that lets your cellular radios work the way it's supposed to. Calling and providing internet service.

THe problems with your Wi-Fi is probably with your internet provider. Because there shouldn't be an issue with your Wi-Fi not working unless it's with your provider. Have you tried using Wi-Fi at other places with your phone.

But if your Wi-Fi doesn't work while other devices do, then it's probably a hardware failure. You should get your phone replaced as soon as possible. Hope this helped.