The+officers+were+satisfied.+My+manner+had+convinced+them.+I+was+singularly+at+ease.+They+sat,+and+while+I+answered+cheerily,+they+chatted+of+familiar+things.

This quote is so tricky in a way. The narrator thinks that he has his crime in the bag and that there is no way he can be caught for his crime. But if you step back and read the story not in the narrators point of view, you can realize that maybe he wasn't as clever as he thought he was. If he really had the police fooled, they would have left a long time ago because they knew nothing is wrong. We know that police officers don't stay unless they need too and that they must have suspected something wrong with the narrator. If you think about it, the police officers knew that the narrator was a weird person and they doubted that it was him screaming from his dream, so they decided to stay awhile and see if time would make him crack.