I think it depends on your area. I am in Texas. For my land line it is about 10 days. For my mobile I have gone as many as 6 days late without it being turned off. If you call them in advance of the due date or termination date and explain how much additional time you need they will usually grant you an extension.
AT&T will turn off the service three days after the date it was due to be paid. You can have your phone left on if you contact the service provider and have an arrangement made. For more information on late payments, check http://mightymom.hubpages.com/hub/The-Softer-Side-of-ATT.