Using finite-difference time-domain simulations, we study the interactions of electromagnetic radiation with a square array of dielectric rods parallel to the electric vector. We observe the electric and magnetic Mie resonances which induce intervals of negative effective permittivity and permeability and which contribute to the formation of the photonic band gaps. Owing to the interplay of these phenomena, a narrow spectral range with a negative refractive index can occur. However, this requires the filling fraction of the dielectric to fall into a well defined interval of values and its permittivity to exceed a minimum of about 50. We discuss these phenomena from the perspective of both photonic crystal and metamaterial concepts.