Except that no one could explain why it happened to my wife's phone. It was not e-mail. When I called AT&T they kept trying to blame it on some app or another. I kept telling them that she did not have any of those. They gave up, but continued to blame it on something on the phone.
MotionMan
It could be anything; push notifications, e-mail, app update notifications, etc.
The problem with iOS is that when a network connection is erratic, the transactions often keep getting retried over and over again. I've noticed this on some versions with e-mail, if you have an unstable connection, messages will keep getting downloaded over and over and over again - the same ones - the download restarting from the beginning if it gets interrupted. Eventually, the mailbox update completes, and duplicate messages get purged and the system calms down. Meanwhile your battery is getting killed and data allowance demolished.
This is more a problem on the iphone than most android devices, as the iphone always prefers cellular data to wifi when the phone is not being used interactively. When the display is off, cellular data is preferred, even if wifi is available.
At least on most android devices, you can configure whether wifi is preferred when the phone is asleep (at the cost of more battery drain). This isn't an option on iOS, unless you turn off cellular data (which I ended up having to do whenever I was at home, because the patchy cellular coverage would cause massive "silent" data consumption, drain the battery, and make all the system apps (like e-mail) go haywire). Diagnosis is easier on android, too, because the OS maintains per-app data counters.