© 2016 American Physical Society. Magnetic properties of thin Co/Pt multilayers have been investigated in order to study the dependence of magnetization M, uniaxial anisotropy Ku, and Curie temperature TC on the multilayer thickness, composition, and structure. A comparison between epitaxial submonolayer multilayers and epitaxial fcc CoPt3 alloy films with large perpendicular magnetic anisotropy (PMA) attributed to growth-induced Co clustering reveals significant differences in the temperature dependence of magnetization M(T), despite the presence of thin planar Co platelets in both cases. Even the thinnest discontinuous multilayered structure shows a Langevin-like M(T), while the alloy films with PMA show a broadened and enhanced M(T) indicating a distribution of environments, including monolayer Co platelets separated by only 1-2 layers of Pt. These differences have been reproduced in Monte Carlo simulations, and are shown to be due to different distributions of Co-Co and Co-Pt nearest neighbors. The relatively uniform Co-Co coordination of even a discontinuous rough multilayer produces a Langevin-like M(T), whereas the broader distribution associated with platelets in the PMA films results in a nearly linear T dependence of M.