What is the motherboard? What is the exact cooler make and model? Is this a Dell pre-built by any chance? A picture would help a lot.
If you are referring to an aftermarket cooler mounted on the board, it is possible the posts you see remaining are from the cooler's backplate which has to be removed. If you can access it from the cut-out in the case's motherboard tray, no problem. If not, you will have to remove the motherboard from the case.