I hiked the CDT and many times in NM wished I had a bike. Lots of road walking. The best parts of the hiking were the Gila and Pecos Wildernesses, and you'd have to detour those.

Do you mean right now? Whether it's possible or not depends on time of year. Good seasons, at least for hiking, are April, parts of May (extreme winds possible), and parts of the fall, I hear. It would probably be suicidal to go out there in winter.

The Arizona Trail has bike detours set up, as an option, probably same time constraints as the CDT.