A six-page (and growing) forum post on Apple's support website finds numerous owners of the Verizon-compatible iPad 2 reporting problems connecting to Verizon's 3G network.
The problem seems to be that if the iPad's Verizon connection is turned off, the tablet must be rebooted in order to get it connected again. This glitch only affects the Verizon connection; the Wi-Fi connection isn't affected.
Users are reporting that calls to Apple and trips to Apple's Genius Bar aren't helping. The only apparent fix is to turn the Verizon connection back on in the settings and then reboot the iPad. One forum poster said that turning on data roaming in the settings has worked as well.
The first forum entry popped up on March 15th and, despite a software update that became available on March 25th, hasn't been addressed yet. There's another update rumored to be coming in the next week or two that'll hopefully patch up Verizon's connection problems.