I had the exact same problem. I went to the AT&T store who told me I need to take it to the Apple Store. They also said that since my phone was more than a year old, I couldn't get a replacement, otherwise Apple would replace it for free. They also said the problem may have something to do with the screen. That's when I suddenly realized I had the screen and the LCD replaced a couple months ago, since my phone fell and the screen broke. So I called the guy who had replaced it, he came over, changed the screen and voila! It worked!
Try replacing the screen. that might work. Good luck!