Here is a better example of the type of player who could get the shaft (although he will likely be eligible due to All-Star game appearances)
http://whatifsports.com/HBD/Pages/Popups/PlayerProfile.aspx?pid=999070. In the middle of his 7th season and it will be closer if he appears in 50 games. If he does it will be his 5th in 7 seasons. In his career he 220 saves in 250 attempts, a 1.05 WHIP, and a 2.61 ERA. His last 5 seasons (including the current ones have been as follows:
37 games, 35 IP, 29/35 Saves, 1.11 WHIP, 3.09 ERA
45 games, 40 IP, 36/41 Saves, 1.35 WHIP, 3.60 ERA
61 games, 61.1 IP, 39/45 Saves, 0.93 WHIP, 2.93 ERA
57 games, 54 IP, 40/45 Saves, 0.89 WHIP, 1.83 ERA
31 games, 27.2 IP, 25/26 Saves, 0.90 WHIP, 0.65 ERA (current season more then half way over)