The convex cone of n×n completely positive (CPP) matrices and its dual cone of copositive matrices arise in several areas of applied mathematics, including optimization. Every CPP matrix is doubly nonnegative (DNN), i.e., positive semidefinite and component-wise nonnegative. Moreover for n less than 5, every DNN matrix is CPP. We investigate the difference between 5×5 DNN and CPP matrices. We give a precise characterization of how a 5×5 DNN matrix that is not CPP differs from a DNN matrix, and use this characterization to show how to separate an extreme DNN matrix that is not CPP from the cone of CPP matrices. Joint work with Sam Burer and Mirjam Duer.