Apple has acknowledged a bug in iOS 6 that prevents scheduled Do Not Disturb from turning off in the morning. The bug was first discovered yesterday when the new year rolled around. Apple has said that the bug will automatically fix itself next week, so until then they recommend manually turning Do Not Disturb off.
"Do Not Disturb scheduling feature will resume normal functionality after January 7, 2013," Apple said in the support article. "Before this date, you should manually turn the Do Not Disturb feature on or off."
Ironically, Apple started the advertisement below around the New Year, in which the Do Not Disturb feature is promoted:
If you use the Do Not Disturb feature and need to make sure calls are coming through, you might consider disabling the feature until after January 7.