In the field of spintronics, ferromagnetic/non-magnetic metallic multilayers are core building blocks for emerging technologies. Resonance experiments using stripline transducers are commonly used to characterize and engineer these stacks for applications. Up to now in these experiments, the influence of eddy currents on the excitation of the dynamics of ferromagnetic magnetization below the skin-depth limit was most often neglected. Here, using a coplanar stripline transducer, we experimentally investigated the broadband ferromagnetic resonance response of NiFe/Cu bilayers a few nanometers thick in the sub-skin-depth regime. Asymmetry in the absorption spectrum gradually built up as the excitation frequency and Cu-layer thickness increased. Most significantly, the sign of the asymmetry depended on the stacking order. Experimental data were consistent with a quantitative analysis considering eddy currents generated in the Cu layers and the subsequent phaseshift of the feedback magnetic field generated by the eddy currents. These results extend our understanding of the impact of eddy currents below the microwave magnetic skin-depth and explain the lineshape asymmetry and phase lags reported in stripline experiments.