Poly(carbon monofluoride), or (CF)n, is a layered fluorinated graphite material consisting of nanosized platelets. Here, we present experimental multidimensional solid-state NMR spectra of (CF)n, supported by density functional theory (DFT) calculations of NMR parameters, which overhauls our understanding of structure and bonding in the material by elucidating many ways in which disorder manifests. We observe strong 19F NMR signals conventionally assigned to elongated or "semi-ionic" C-F bonds and find that these signals are in fact due to domains where the framework locally adopts boat-like cyclohexane conformations. We calculate that C-F bonds are weakened but are not elongated by this conformational disorder. Exchange NMR suggests that conformational disorder avoids platelet edges. We also use a new J-resolved NMR method for disordered solids, which provides molecular-level resolution of highly fluorinated edge states. The strings of consecutive difluoromethylene groups at edges are relatively mobile. Topologically distinct edge features, including zigzag edges, crenellated edges, and coves, are resolved in our samples by solid-state NMR. Disorder should be controllable in a manner dependent on synthesis, affording new opportunities for tuning the properties of graphite fluorides.