Some users changed the date as an emergency fix, but it must have been annoying for those waiting for a call just after the clock ticked 12. What Do Not Disturb does is switch the phone into silent mode so it does not ring or vibrate. An exception to that is alarms set in the native clock application.
So if someone you’ve wanted to call doesn’t answer, don’t panic - it is probably their phone playing tricks on them.
Ironically, Apple just released its new iPhone ad explaining how awesome the DND feature is. Not the best timing.