If you just got a Verizon iPhone and are having trouble getting your contacts transferred, make sure you turn Wi-Fi off before starting the VZ Contact Transfer app. This forces the VZ Transfer app to use your Verizon 3G connection to transfer your contacts.
To turn Wi-Fi off, tap the Settings app on your iPhone and tap the Wi-Fi setting. Once it's off, your status bar will switch from showing the wifi signal icon to "3G".
Now tap the VZ Transfer app and try again.
The error you'll see if you don't do this is:
Network Error Application lost network connection. Please retry your Operation.
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