Most definitely. Hart effectively went out on a high note after helping to keep us up under Slade. Pretty much everyone knew that was his last hurrah. After that season he was always expected to slowly start taking a backseat. Adam on the other-hand won't get anywhere near the same sign-off...
Even if Barnsley are - as they were in the summer - I wouldn't be surprised if Bristol City could offer Adam more money (and more money to The Albion for that matter).