Like many substances, carbon fiber can be recycled, but it may (currently) be better described as downcycling, vs. recycling.
Carbon fiber composite materials, for example, are being used increasingly in aircraft manufacture. Although this has been the case in more expensive military aircraft for decades, only recently have large commercial aircraft started using more composite materials than aluminum (the new Boeing 787 is about 50% composites vs. 20% aluminum).
From this article about Boeing
... use a process called pyrolysis, whereby heat removes the resin
from shredded raw composite material, rendering what Carberry
described as a kind of fluffy fiber.
Carberry explained that physical properties of recycled composites
limit their use to nonstructural parts. Only new material–made of
virgin carbon fiber–is strong enough for fabrication of
The carbon composites are heated to around 400 °C - 600 °C, so the epoxy resin can be burned off (and lost), and the fibers reclaimed. Although the end result is not a fiber of the same length as the original, the process only uses about 5% of the energy that the original manufacturing process did.
Like when paper is recycled, the resulting fibers get shorter and shorter, and this reduces the strength of the recycled composite material. They are still usable, but not for all the same applications (e.g. aircraft structure) as the original material.
An early, lower-tech recycling process for carbon fiber simply involved grinding it up, and using it as filler material for applications like road building. Less demanding applications like laptop cases are also current recipients of recycled carbon fiber.
Generally, the material performance demands are highest for aerospace applications, then automotive applications, then sporting goods, and finally consumer electronics. So, this is likely to be the downcycling flow for carbon fiber (although I don't have evidence that recycled aerospace fibers are being used in car frames yet). It's also the case that aerospace consumption of carbon fiber composites is vastly greater than smaller consumer products, so that's the most important place to start innovating in fiber recycling.
Check out this article, though, on Trek Bicycles recycling carbon fiber from bike frames
Here is a link for a company developing a different carbon fiber recycling process
Another link discussing the general state of the art, and future directions
Disclaimer: in a former life, was a Boeing structures engineer, and did my Master's in Aero/Astro structures.
Edit: Please also be aware that carbon fiber sustains strain and damage during use and cannot be 100% recycled.