Do you have the line filters installed on all of the telephones? If you don't, then picking up the phone substantially alters the characteristics of the line as far as DSL is concerned.
Experiment: Disconnect all of the telephones in the house and see if the internet connection works. That also gives you a chance to verify that filters are on for every phone (It is also possible one of the line filters is bad). If that works, then reconnect the telephones one at a time to see which one is causing the problem.
If it doesn't work with none of the telephones connected, get a Verizon tech out to sort it out.
