Perhaps you've erased the move from your memory as a defense mechanism of sorts, but the Mets once acquired Luis Ayala. In the middle of August 2008, the team traded Anderson Hernandez to the Nationals for Ayala as the Mets attempted to stay in the playoff race coming off the "non-devastating" 2007 season.
At the time, Ayala had made 62 appearances for Washington and had a 5.77 ERA. Perhaps the Mets thought Ayala could turn back into the 2.82 ERA pitcher that he was over his first four seasons in Major League Baseball, but he didn't improve much after the trade. With a 5.50 ERA in 19 appearances for the Mets, Ayala certainly didn't help the team as it missed the postseason by one game for the second year in a row.
From there, Ayala hit free agency. He wasn't good in 2009 for the Twins and Marlins. In 2010, he roamed the Pacific Coast League, pitching for the affiliates of the Dodgers, Rockies, and Diamondbacks. With a 6.42 ERA against Triple-A hitters, it looked like Ayala's big league career was over.
In 2011, however, he caught on with the Yankees and posted career bests with a 2.09 ERA and 207 ERA+ in 52 appearances. He's since pitched for the Orioles and Braves, and over the last three seasons, he has a 2.58 ERA and 168 ERA+ with 6.1 strikeouts and 2.8 walks per nine innings. He'll turn 36 in a little over a week, but according to Chris Cotillo, he was seeking a multi-year deal at the end of November, and multiple teams were at least interested in him.
Nineteen games of poor performance for the Mets six years ago shouldn't be a deterrent, but for a team that's built upon the promise of its young arms—in the rotation and the bullpen—signing any relief pitcher over 35 to a multi-year deal does not seem likely.