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I also have Verizon as my carrier. My Ipad just dropped to 3G and my iPhone is still has a strong LTE signal. It must be Verizon.
When it switched, I pressed the 'Sleep' and the 'Home' button and held them until the device powered off. This took about 10 seconds. When I powered on, it would connect to LTE and has stayed for the past three days. While searching this issue, I read somewhere where you could reset subscriber settings. I did that and the problem reappeared. I did my two finger salute and its back to 'normal'. I've several times turned off cellular via settings and then back on, and it's solid LTE.
iOS 6.1.2
Verizon
64gb
iPad 3
Hope this helps someone else.