Cliff Lee, 2010 (twice)
The Yankees somehow managed to botch the Cliff Lee situation twice, coming perilously close to the Cliff's Edge without closing the deal. Sure, New York did capture the 2009 World Series, but they plainly needed Lee to push them over the top into potential dynastic territory in the waning years of the Core Four, unable to trust AJ Burnett and infuse their rotation with talent beyond CC Sabathia/Andy Pettitte.
The Phillies traded Lee to Seattle after the '09 World Series, but despite getting an excellent first half from their new ace, the water-treading Mariners made him available at the deadline once more. The Yankees' deal was ~RUMORED TO BE~ done, featuring Jesus Montero as its headliner, alongside righty Zach McAllister and infielder David Adams. Unfortunately, a physical flagged a foot issue in Adams, and the Yankees were unwilling to replace him with Eduardo Nuñez.
Their loss! Lee instead went to the Texas Rangers for a Justin Smoak-centered package and buzzsawed the Yanks in the 2010 playoffs, much the same way he had in '09. Texas advanced to the World Series in large part because of Lee's 16 innings of two-run ball against the Rays and his eight-inning shutout with 13 Ks against the Yanks, defeating Pettitte 8-0.
At least the Yankees were still the favorites to sign the rental after the season! Right? ...Right?
Eh, no. Lee went back to the Phillies, and spurned the Yankees amid an unpleasant storyline that fans had spit on his wife during the Rangers ALCS, which might've been the most pivotal postseason series loss of the era, non-Derek Jeter's ankle division.