Unless you're carrying around a very old device, your phone has a specified capability* that makes sure official POTUS messages can always go through, no matter what. Phone manufacturers are required to include technology to enable these communications in devices, due an extension of the Emergency Alert Network enacted in 2012 that allows the president to get in contact with anyone, anywhere in the US, whenever it's necessary.
The emergency system allows phones to receive presidentail emergency alerts when they enter a specific geographic area. Technically, it's not actually a text. (So you'll never see Sent From my BlackBerry on emergency warnings from the president.) It's a specialized message form developed for emergencies.
These warnings have a unique vibration and sound alert, and they're sent to any phone within a geographic radius of a cell tower. Unlike Amber Alerts and weather emergency alerts, there is no way to opt out of the presidential text. It's compulsory.
Oddly enough, if you do get a presidential emergency alert and you're on the phone, it won't show up until after you've finished your conversation. I guess the apocalypse can wait?
*Correction: Originally I referred to the deployment system as a special chip, which is not the case. Thanks to commenters for pointing that out.
Image by Whitehouse/Pete Souza