Siem de Jong was left unchecked at two first-half corners, the Ajax captain able to head home a brace, forcing Yaya Toure and Sergio Aguero to dig deep and score the goals needed to keep City within five points of the qualification spots in Group D.
Kompany told ITV1: "It's very important to defend as a team when corners come in and we've probably let ourselves down on those two occasions.
"Those two goals were really not good.
"We'll look back at the game, try to see what went wrong, and try to improve.
"It's not the case that we're not good enough for the Champions League but it's a matter of time.
"It's unfortunate that we're letting those chances go at the moment."