I am having the same issue and for a Verizon Support rep to suggest a hard reset after reading all these messages is crazy. I am on my 4th week now of troubleshooting, my 3rd replacement phone, running no apps, at least 6 security wipes, multiple times of re-registering and re-sending service books, and still having the same issue.
My wife and I have the same phones and I upgraded both to 6.0 on the same day and that is when it started. I have been a Blackberry user for 5-6 years and never had this issue until 6.0. When they send me a replacement phone it works for 3-5 days with maybe having to do a battery pull once. Then, the gremlins come back. Not only does it do the rebooting thing, but they will stop working 3-4 times a day where you cannot make a call, or use the data for anything (web, e-mail, etc.), nor will it send or receive texts.
I have called in approximately 10 times to tech support and everyone is very nice and the bump me up to blackberry support right away. They don't have any other options than to replace your phone with something other than a blackberry. My wife has had it and now is disgusted with Blackberry. I still love them and wish they would resolve the issue. They have shipped us a Droid for my wife and I was having luck with my last replacement for about 5 days but now it is doing the same thing again. It has to be something to do with the operating system and how it reacts to the network signals it is picking up on. That is my guess. They need to get their top Blackberry Wizard (from Blackberry, not Verizon) to dring a bunch of Red Bull or whatever and get his butt to work on this problem until it is fixed.
This is costing me money in my business when I cannot receive an e-mail and respond to it ASAP. Someone, PLEASE fix this problem and let Blackberry know that they are about to lose a lot of customers.