You don't say "how" they fail.
The most common issue in these types of situations is a malfunctioning DNS server somewhere.
As a test, try changing the DNS server on your PC to use an alternate DNS service. Google (8.8.8.8) or OpenDNS (208.67.222.222 and 208.67.220.220) are two high availability / reliable services. This will isolate if the issue is with name resolution using the Verizon servers or somewhere else.
Additionally, pick a site which won't load and open a command window on your PC and do a "tracert" to the hostname (tracert www.google.com) and post the results.