The "Reset Network Settings" button on the iPhone not only resets your Wi-Fi settings -- resulting in a clean slate with respect to Wi-Fi access points and their passwords -- it also resets your phone carrier settings. Several months back, I had the darndest trouble with cellular data access. I could place and receive calls, but couldn't get cellular data access no matter what I did. I tried going into and out of airplane mode, turning the phone on and off, and resetting it, all with no luck.
So I decided to call AT&T. Apparently, one of the cell phone towers near my office was having trouble and fell out of and back into commission. And although the tower was back up and running, my iPhone was stuck in cellular data purgatory. AT&T support suggested that resetting the network settings on the device would likely resolve the issue: and it did. Resetting network settings has also helped me resolve an issue I was having with Visual Voicemail on another occasion.
To reset your iPhone's network settings, tap your way to Settings > General > Reset > Reset Network Settings. With great power also comes great annoyances, like having to reconnect to previously remembered access points and having to re-enter passwords.