It would probably work for one firing just fine. But for this application I can see several problems.
Carbon fiber is so stiff (modulus of elasticity is ~2.3 Gpa, compared to steel at ~2.0 Gpa and Brass at ~1.0 Gpa) is would likely not obturate well to seal the chamber, especially in lower pressure cartridges. Steel case suffers this to a lesser degree.
Carbon fiber typically fails in a brittle manner, where soft steel, aluminum, and brass currently used in cartridges fails in a ductile manner. This failure mode means when carbon fiber when if fails, such as a case head failure, it would create more high velocity fragment than soft steel or brass.
It would probably not reload well since it would not really resize well as it have very little plastic deformation before permanent damaged. The act of firing it would ablate some of the resins used to hold the fibers together on the inside surface again causing problems for reloading.
Making the cartridge would be costly. If you simply take a bar of carbon fiber and cut a cartridge shape out of it (straight wall would be doable) you have cut much of the carbon fibers weakening the case significantly. You would need to really lay up the cartridge on a mandrel so you reach very close to net final shape with little or not machining to cut as few fibers as possible. This would be very hard/expensive to do for bottle neck cartridges.
It's all doable but I don't see much advantage. It's also not really needed with cases like True Velocity coming along quite nicely.
-rambling.