We had basically the same experience with AT&T (which in our area was bought by Comcast and then by Time Warner). Slow speeds, old landlines that would go down in the rain, and service calls that accomplished nothing.
Then Verizon installed a fiber optic (FIOS) network in our neighborhood. I paid extra for 20 mb service, and regularly clock speeds of 25 mb or higher. I got a good rate on an TV/Phone/Internet package. Fantastic service. I laugh now when AT&T/Comcast/Time Warner advertises Internet on TV. No way I'm ever going back.
What do you pay for your Verizon FIOS setup?
We don’t have them yet in our neighborhood.