It looks like there is a bug in the Verizon version of the iPad 2, where the cellular radio will not turn on if the system was previously powered off or reset with the Cellular Data switch in the off position.
In order to start the radio, you need to set the Cellular Data switch to the "on" position, then either shut down and power up, or reset the iPad (press and hold the Sleep/Wake button and the Home button at the same time until the Apple logo appears).
I called Apple, and at first, the CSR had no knowledge of the issue, but after putting me on hold and checking with others, he came back on the line and said that they were working on a patch.
Hope that this is helpful to anyone else experiencing this issue.
This is not known to occur in the AT&T version.