John Carney had an incredible year as the Giants’ place kicker in the 2008 season. Why didn’t he get re-signed? Too old?
Kickers are, for the most part, interchangeable.
He was just brought in to hold down the fort when Lawrence Tynes – who helped them win the Superbowl the year before – went on IR.
FG % is a fickle thing that isn’t particularly consistent from season to season. Kickoff distance, OTOH, is consistent, and Carney was consistently bad. Tynes isn’t great at kickoffs, but clearly better than Carney. Plus he’s younger and also had that “helped you win the Superbowl” thing going for him.
EDIT: It’s really more a matter of Tynes helped them get to the Superbowl by kicking the game winner in Green Bay. The fact that he missed two different game winners just before that isn’t a strike against him only because of that game against the Seahawks in 2005 when Jay Feely missed three game winners. So when Tynes actually hit the third one he must have seemed like a savior for the Giants, who had been burned by choking kickers for the better part of the decade.
As Ellis says, kickoffs played a big part. Another aspect was that Tynes gives them the option to try longer field goals. Carney was/is accurate, but the best you can hope for 45 yards on a good day. Carney’s kicks won’t get any longer, but Tynes can hopefully get more accurate.
Yes, too old. He’ll be 47 soon. Too old even for a kicker. That is the reason. And, I suppose, all the great football reasons others have outlined factored in the decision; but a lot of that is due to his age.
But wasn’t Morton Anderson 56 when he retired from the Falcons?
Jesus, was he!? But certainly that is rare. Morton Andersen was an exceptional kicker, unequaled. A bit of reasearch tells me he is now only 50, so I don’t think so. But if he did kick till 56, it wouldn’t surprise me; however, kickers of his ability are so rare.
Carney, of course, wound up getting brought in by the Saints in '09 and '10 when Garrett Hartley got injured or was ineffective.
He’s not retired. He’s still hoping to catch on with somebody. 
But, seriously, as others have noted the issue with Carney (and Morten Andersen, and Gary Anderson, all of whom kicked well into their 40s) is that, while they’re highly accurate, they wind up losing leg strength, which limits how teams can use them – they can’t count on 50-yard field goals (which most teams consider to be a minimum requirement for a placekicker today), and their shorter kickoffs mean that either the team is giving up field position, or they need to use someone else for kickoffs (often the punter).