I am suffering the same issue, and restoring the device didn't solve it. As the issue is happening with preinstalled Apple apps: Mail, Weather,... i think we cannot do so much. It is a iOS software bug, which needs to be addressed by Apple, and it is not a Hardware problem.
I have been web researching and this issue began on the iOS3.x. Basically, you can turn off the cellular data, and even delete its APN configuration. But by some strange reason (iOS BUG), when you use one of the native apps as Mail, Weather, etc., the iPhone will try to establish a cellular data connection. Of course, if the APN configuration has been deleted, the iPhone will not be able to establish the data connection, but if your operator configures its Mobile Network to bill you those failing data connections "tryouts", you will be billed for them. Basically, is like paying by making "lost calls".
It is not fair, of course, but the problem is not the operator, the problem is that the iPhone doesnt fully disable the cellular data connection, even turning OFF all the settings.
This is a MAJOR bug, and from my research, the only solution that I found was mentioned in a Jailbreak forum. Of course, this is not legal, because we have not our iPhones jailbreaken, and we want that Apple solves this iOS BUG definitely.
This is an iOS bug since 2010 at least, unbelievable, but true. 😐
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