The aim of this work is to analyze the role of the impressed sources in determining the well or ill-posedness of time harmonic electromagnetic boundary value problems involving isotropic effective media. It is shown, in particular, that, even if all interfaces are regular, the class of ill-posed problems can be very large in the presence of general square-integrable impressed sources. However, when a simple and realistic constraint is enforced on these sources, requiring that the support of the sources does not include any interface between a traditional medium and a metamaterial, among the problems here considered just those involving an interface between complementary materials remain ill-posed. These considerations have a very significant impact also on the approximability of the solution of well-posed problems since the numerical noise can introduce small fictitious sources even where the sources to be simulated are not present. These effects on finite element simulators are fully analyzed. Finally, we propose an algorithm that allows to obtain much better approximations of the solutions of the most critical well-posed problems.