Bradley Cooper does not have an Academy Award because he simply has never ended up winning one or an Oscar. He is good enough to have both. An Academy Award is not like an Oscar in that you don't really know who has been nominated for one and who has not been nominated for one. It is more secretive and suspenceful than an Oscar nomination is.
One thing that's for sure is that Bradley Cooper is a great actor. Many have felt sorry for him in the past for not winning this or that which is embarrassing to him. He would rather just go about his acting and directing that to have pity. He's a good man and a very professional person so he will be fine as long as his fans and the public will allow him to be.