A single chromophore can only emit a maximum of one single photon per excitation cycle. This limitation results in a phenomenon commonly referred to as photon antibunching (pAB). When multiple chromophores contribute to the fluorescence measured, the degree of pAB has been used as a metric to "count" the number of chromophores. But the fact that chromophores can switch randomly between bright and dark states also impacts pAB and can lead to incorrect chromophore numbers being determined from pAB measurements. By both simulations and experiment, we demonstrate how pAB is affected by independent and collective chromophore blinking, enabling us to formulate universal guidelines for correct interpretation of pAB measurements. We use DNA-origami nanostructures to design multichromophoric model systems that exhibit either independent or collective chromophore blinking. Two approaches are presented that can distinguish experimentally between these two blinking mechanisms. The first one utilizes the different excitation intensity dependence on the blinking mechanisms. The second approach exploits the fact that collective blinking implies energy transfer to a quenching moiety, which is a time-dependent process. In pulsed-excitation experiments, the degree of collective blinking can therefore be altered by time gating the fluorescence photon stream, enabling us to extract the energy-transfer rate to a quencher. The ability to distinguish between different blinking mechanisms is valuable in materials science, such as for multichromophoric nanoparticles like conjugated-polymer chains as well as in biophysics, for example, for quantitative analysis of protein assemblies by counting chromophores.