Try updating your PRL (list of towers) by dialing *228 and choosing option 2. If that doesn't work, you can also try reprogramming your phone by dialing *228 and choosing option 1.
Have you checked to be sure that your phone's memory is not over-full? That may be preventing some or all of your messages from being delivered.
If reseting the phone doesn't work, you may need to call Verizon's tech support. Just dial their 800 number from another phone and choose option 3. (Using another phone to call makes it easier for them to go through troubleshooting steps with you.)