Exchange-coupled nonmagnetic metal (NM) and ferromagnetic metal (FM) multilayers are crucial for microwave magnonic and spintronic devices. These layered materials usually have total thicknesses smaller than the microwave skin depth. By using a stripline broadband ferromagnetic resonance spectroscopy technique, we experimentally demonstrate that the amplitude of the magnetization precession in the FM layer is strongly diminished by the shielding effect of microwave (6-12 GHz) eddy currents circulating in the NM capping layers.