The Premier League champions were unfortunate to be eliminated from the competition at the group stage last season after amassing 10 points, but went out of this season's iteration with a whimper - picking up just three points and finishing bottom of the group.
Tuesday night's defeat to Dortmund left Hart furious, with the goalkeeper upset with his side's reaction to going a goal behind in the second half.
When asked if City have let themselves down in Europe, Hart told Sky Sports: "Yeah, of course we have. We're out of Europe, we're not even in the Europa League so that's fair to say.
"There was a hunger to win, like I say we've let ourselves down in this tournament but there was a hunger to win.
"If that meant the Europa League then great, but we've got bigger issues that that, we need to be winning playing [against] a weakened team tonight.
"Congratulations to Borussia and Real Madrid I thought they were fantastic, but we should be fighting, definitely fighting last day to be in the competition and we weren't.
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"[Intensity] was there and then they scored, and it disappeared which is a shame to be honest because we're a side who can come from behind. We've proved it in both competitions so it was disappointing."
The England No.1 dismissed the notion that the defeat would affect City's performance against Manchester United on Sunday, and is itching to set the record straight in Europe next season already.
He added: "We should have been in that position [to qualify] and we weren't, we only have ourselves to blame, that's it.
"We've disappointed in Europe and been fantastic in the league, we've been consistent and won so I don't think there's any worries about that but we do need to address it for getting in the Champions League again, winning the league and hopefully we'll be able to deal with it next year."