The Verio question is confusing the difference between network and backbone. Verio owns its own backbone.
Running your own network is different. It just means that the servers are connected via a network that is owned by the host. This could be something very simple, or quite complex really. Typically you would have a router between you and your upstream network provider.
Also, Dinix definately does run their own network which connects directly with InterNAP.
Originally posted by Joshua Dinix is directly connected to Internap, and has a direct contract with them, and doesn't buy bandwidth or rent servers from anyone else.
Dinix has their own cage/vault at e^Deltacom. While they are "directly connected to Internap," one big thing about their move was that they brought a couple extra providers into BGP just for failover -- I've never seen traffic flowing to/from my VPS on anything but Internap, but if for some reason their whole Internap feed went down, you're not dead in the water. (But as long as Internap's up, you're on it.)
It's not clear whether you wanted a place with their own datacenter and their own nationwide network, or just someone with their own cage and all like Dinix, but I wouldn't hesitate to recommend them.