The incoherent dynamical properties of open quantum systems are generically attributed to an ongoing correlation between the system and its environment. Here, we propose a novel way to assess the nature of these system-environment correlations by examining the system dynamics alone. Our approach is based on the possibility or impossibility to simulate open system dynamics with Hamiltonian ensembles. As we show, such (im)possibility to simulate is closely linked to the system-environment correlations. We thus define the nonclassicality of open system dynamics in terms of the nonexistence of a Hamiltonian-ensemble simulation. This classifies any nonunital open system dynamics as nonclassical. We give examples for open system dynamics that are unital and classical, as well as unital and nonclassical. Ref: H. B. Chen, C. Gneiting, P. Y. Lo, Y. N. Chen, F. Nori, arXiv: 1703.09428, Phys. Rev. Lett. in press (2018).