We investigate the magnetic field and current distributions in structures made of superposed superconducting films. A recent experimental study of a planar bilayer structure (Tamegai 2017) has revealed unusual patterns in the critical state, with an unexpected discontinuity line arising in the region where the layers overlap. We use a finite element model to investigate the critical states in such 3D structures with several layers and different overlapping configurations. We show that the experimentally observed discontinuity line arises as a result of a magnetic field dependence of the critical current density. Additionally, we show that the network of discontinuity line not only depends on the geometry of the system but also evolves with the applied magnetic field as the structures are magnetized in a zero-field-cooled mode. The strength of the magnetic coupling is shown to depend on the number of layers, their overlap and separation, with a direct effect on the length and shape of the discontinuity lines.