I rode a section here in New Mexico last fall through what is called the Chain of Craters area. The challenge was that the trail itself disintegrated after a while. Then I was overlanding and following cairns. But at some point, I couldn't locate the next one so I had to double back and jump on the forest road that roughly parallels it. Other sections of the CDT here are actually on forest roads (like up through Bonita Canyon where I have also overnighted). And then there are parts that are single track and not in wilderness areas. So, a lot of factors in figuring out which sections are ride-able. Get a hold of some good maps, I say, and see what things look like on aerials as well. That's how I plan most of my trips. I find where a trail or road is supposed to be and what is actually there on the ground can be quite different, so it pays to study the aerials before heading out.
Here is link to the Chain of Craters Back Country Byway trip I did last fall (that included some CDT and a not very well work singletrack):
http://highaltitudehomestead.blogspot.com/2013/03/bikepacking-chain-of-craters.html