Incurring Text Charges While Out of the Country

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@genedirectv wrote:

My wife and I were in Mexico last month with our iPhones set to "Cellular Data" OFF prevents any data use.

, "Roaming" OFF,  prevents data roaming, and Iphone can prevent calls while roaming IF you also set that to off.

and "Wi-Fi Calling" ON.  If you have a Wifi connection, calls and text are free to the US.  Calls to Mexican numbers/non US numbers are charged as long distance.  
My wife has the international plan were if you make at least one international call (which they all will be since we were out of the country) you get charged $10 for the day.  We assumed all our texts were going through Wi-Fi as the resort had Wi-Fi just about everywhere.  From what I understand AT&T's system would only pick up the message if it was on cellular.  But how can that be?  If we were set to Cellular OFF how did that happen?  Could some other app switch the "Cellular Data" to ON?   Thanks in advance.

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None of those settings prevent calls or text over cellular.  

Hotel and resort Wifi is notorious for disconnecting while idle.   The correct method to prevent cell connection is airplane mode, then turn Wifi back on.   Then you would know when you were disconnected from Wifi.