Yep - we always turn our phones off when we get on the plane to fly to Mexico. Before turning them off, we turn off ALL Cellular Data and then set for WIFI only so that if and when we turn them on, we do not have all our emails or text messages loading at data roaming charges. I learned that on TUG and have also read the AT&T information about international travel. We use Skype either from our laptop or via WIFI on our cell phone to call back to the USA. Cellular Roaming charges in Mexico can really add up! Be sure to turn off all the tracking apps too. None of your apps need to know where you are.
Last year, we drove to Mazatlan for a month and when we crossed the border into the US and were waiting for about an hour to get through the Customs area, I turned my phone on and emails started loading and I also made a call to let our "kids" know that we were back in the USA. That caused a $20+ charge on our bill. I called and contested it when I got the bill because I was in the USA before I turned the phone on. The gal suggested that in the future that I wait a while before turning my phones on in the future, but she did give us credit for the charges. This year I will wait until we stop for dinner somewhere between Nogales and Tucson.