Why do bodies age if their cells are constantly replaced?

Dear Cecil, Straight Dope Science Advisory Board, or Teeming Millionth:

I’ve heard that every cell in my body eventually dies and is replaced by another cell. This “serve, die, and get replaced” cycle will continue until, eep, my death. Supposedly I could say that my entire body is comprised of 100% different material after “x” months. (“x” being a number that I don’t remember - but could be, for argument’s sake, eleven) If that’s true, then why do: my joints stiffen, my scars stay forever, and other yucky aging things happen as I get older? It seems to me that all these vibrant, springy new cells have no right making me saggy and sluggish.