When renting a dedicated server or a box for community purposes, will the network carriers most hosts use affect how fast or slow your game server gets players?

nLayer is our choice, it has proven EXCELLENT.

For the record, we also use Level3 and Internap in some of our locations, but we prefer nLayer when available.

Yes, the better the network the better the ping performance. nLayer, InterNap, and Level3 are great for game servers.

I am sure Level3 is one of the better ones around :)

Level3 definitely doesn't beat all :). It's not bad, but a good tier 2 can easily beat it. Mzima, InterNAP, or nLayer are generally going to perform better than straight Level3, and they'll offer the additional benefit of better redundancy, as well.