When a coronal mass ejection (CME) appears in a coronagraph it often exhibits three parts. This "classic" threepart configuration consists of a bright leading edge, a dark circular-or teardrop-shaped cavity, and a bright core within the cavity. It is generally accepted that these are manifestations of coronal plasma pileup, the driving magnetic flux rope, and the associated eruptive filament, respectively. The latter has become accepted by the community since coronagraph CMEs have been commonly associated with eruptive filaments for over 40 years. In this second part of our series challenging views on CMEs, we present the case that the inner core of the three-part coronagraph CME may not be, and in the most common cases is not, a filament. We present our case in the form of four exhibits showing that most of the CMEs in a broad survey are not associated with an eruptive filament at the Sun, and that the cores of those CMEs that are filament-associated do not geometrically resemble or consist of material from the associated filament. We conclude with a discussion on the possible causes of the bright CME core and what happens to the filament material postlaunch. We discuss how the CME core could arise spontaneously from the eruption of a flux rope from the Sun, or could be the result of a mathematical caustic produced by the geometric projection of a twisted flux rope.