In model checking, the state explosion problem occurs when one verifies a non-flat system, i.e. a system described implicitly as a synchronized product of elementary subsystems. In this paper, we investigate the complexity of a wide variety of model checking problems for non-flat systems under the light of parameterized complexity, taking the number of synchronized components as a parameter. We provide precise complexity measures (in the parameterized sense) for most of the problems we investigate, and evidence that the results are robust.