The RaboDirect PRO12 outfit were semi-finalists in the competition last season, but have now suffered three straight losses with the latest a 19-9 reverse against Racing Metro in France.
Greig Laidlaw's three second-half penalties were the first points of Edinburgh's European campaign this term after blanks in heavy defeats to Saracens and Munster, but it proved in vain at the Stade Yves du Manoir, Juan Imhoff and Ben Arous scoring crucial home tries.
And with their hopes of qualification from Pool 1 now all but over, Bradley was left to rue missed opportunities.
"We're disappointed with the result because we came here to win and we didn't," he said.
"We haven't performed to the levels and standards we set ourselves in the first two games of this Heineken Cup season, but this was totally different and we were in the game right to the last minute.
"The boys played well and the frustrating thing is that we had opportunities throughout the game and I felt that if we could have scored then, the game would have swung our way.
"But to give Racing their credit, they produced a strong defensive show in that 20-minute period to hold us out, which was a crucial part of the match.
"Unfortunately they then scored a try with probably their only real shot at our line in the second half, and that was it. On another day, it could all have been very different."
Bradley's men will get the chance of immediate revenge to keep their slim qualification hopes alive in the return fixture at Murrayfield next Friday.
And Bradley is urging another display full of determination.
"Despite what happened in previous weeks, we did not come here with the attitude just to limit what Racing could do with the players they have," he added.
"We came here to win but we will take the positives from the attitude the players showed and we are looking to beat them next week at Murrayfield."
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