The Cumbrians entered their final game with a slim chance of reaching the play-offs but, in the end, results elsewhere meant even a victory at Boundary Park would have seen them fall short.
Oldham ended an eight-game winless streak with James Tarkowski's first league goal and an Oumare Tounkara header sealing the points either side of Tom Taiwo's strike.
Abbott said: "I'm extremely proud of the way we keep progressing. We've got obstacles staring us in the face but we got to the 46th game with something to play for.
"Our chance of finishing in the play-offs had probably gone before (this match) but it feels raw at the moment.
"I'm massively hungry to take the club and myself further up the league and I won't be happy until I have us in the top six. It is upsetting that we aren't in the play-offs but when everybody else is winding down I'll be winding up again and starting to plan."