No compelling cosmological models come out of magnetic universes which are based on nonlinear electrodynamics
Here we investigate the cosmic dynamics of Friedmann-Robertson-Walker universes -- flat spatial sections -- which are driven by nonlinear electrodynamics (NLED) Lagrangians. We pay special attention to the check of the sign of the square sound speed since, whenever the latter quantity is negative, the corresponding cosmological model is classically unstable against small perturbations of the background energy density. Besides, based on causality arguments, one has to require that the mentioned small perturbations of the background should propagate at most at the local speed of light. We also look for the occurrence of curvature singularities. Our results indicate that several cosmological models which are based in known NLED Lagrangians, either are plagued by curvature singularities of the sudden and/or big rip type, or are violently unstable against small perturbations of the cosmological background -- due to negative sign of the square sound speed -- or both. In addition, causality issues associated with superluminal propagation of the background perturbations may also arise.