This has to be a Verizon network issue and not an iPhone 5 issue. The wrong date and time bug is happening on my iPhone 5 and my wife's Galaxy S3 for the past 3 or 4 days. We are just on the edge of 4G LTE coverage. It always happens when we are at home and happens when I am on wireless with cellular data turned off and when I have both wireless and cellular data turned off. I don't know if that has anything to do with it or not. On my iPhone 5 toggling airplane mode on and off fixes the issues. On my wife's GSIII it seems to spontaneously correct.
Agreed, this is happening on both of our Galaxy S3's. We've had the phones for a couple months, the problem started about 2 weeks ago. It also seems to happen to us mostly at home, and in other spotty coverage areas, never at work.
I'm tired of my calendar events and alarms going off randomly in the middle of the night because of this, so I've turned off 'automatic time updates from the network' until Verizon fixes the problem.