The Santiago Bernabeu side was dethroned as Spanish champions by Barcelona earlier this term, while it was eliminated from the Champions League by Borussia Dortmund, and the former Valencia man says the club missed the chance to make its season a success on Friday.
"Finishing without any trophies is a failure for us because we are Madrid. We are not happy. Now we have to lift our heads and start thinking about the future," Albiol was quoted as saying on the club's official website.
"Getting knocked out of the Champions League was very painful, but it is no excuse for not winning the final of the Copa. We should have won and dedicated it to the fans. We must not make excuses. We would have liked the season to have turned out better."
Meanwhile, Diego Lopez was equally disappointed with Friday's defeat and felt Madrid was unfortunate not to come out victorious.
"We were unlucky. We had scoring opportunities throughout the match, but luck wasn't on our side. We hit the post three times, clear opportunities at goal. Eight or nine times out of 10 our play today would be enough to win," Lopez explained.
"It always hurts to lose, but more when it's like this. It was our last opportunity to win something this year and the squad was very united for the cause."