Just to flip the scenario a little bit, I can name a fellow that the Orioles thought was past his prime, only to find out, after they'd let him become a free agent, that he had more than a little bit left in the tank.
In 1993, when the Orioles picked him up, this guy had already pitched six seasons, though he had not pitched in the big leagues in 1992.
He went 25-22 over three seasons for the Orioles, which wasn't too bad, but the ERA went from 3.43 to 4.77 to 5.21. At that point, his career record was an unremarkable 59-76 and he was going to turn 33 over the offseason. The Orioles decided he was pretty much at the end of the line and let him go.
He was Jamie Moyer. He would go on to pitch another 16 seasons in the big leagues, winning another 210 games over that time period. His career record was 269-209. He is currently ninth on the list of most games won in a career by left-handed pitchers.