Yeah, it's absolutely possible. Carr/Pitino have to file a waiver to the NCAA for eligibility due to his previous coach being fired. It's similar to the medical hardship/redshirt process. The NCAA also has to change that bi-law, which was one of several under discussion.
I have no idea on a timetable, but I would guess it would be resolved by the fall when school starts. Carr would want to know by then if he's eligible to play. He can still practice with the team, but he's no different than a standard transfer for now until told otherwise.
I can't recall any waiver ever for a fired coach. My bet is he is eligible if they change the rule, not because they make an exception to the existing rule. If that happens, a pretty big group of players would all get declared eligible at the same time. I would say Carr's chances of immediate eligibility are probably below 2%.