We report a time-resolved study of the photoluminescence of CdSe colloidal nanoplatelets with two different thicknesses. By studying the exciton recombination dynamics we assess the exciton fine structure in these systems. The splitting between bright and dark excitons is enhanced compared to epitaxial quantum well structures as result of dielectric confinement. Despite of strong variations in the absolute magnitude, by comparison with literature data we find a relatively slightly varying bright-dark exciton lifetime ratio in very different CdSe-based colloidal nanostructures, regardless of growth technique and of core and shell properties such as materials, dimensions, etc. This finding points to a universal mechanism in the dark exciton recombination.