In this paper, we experimentally investigate the phenomenon of electromagnetic wave tunneling through ϵ-negative (ENG) metamaterial layers surrounded by double-positive layers. Initial experiments are conducted by using a rectangular waveguide, which operates below its cutoff frequency to emulate an ENG layer. This ENG layer is then sandwiched by two dielectric substrates with relatively high dielectric constants and it is shown that the entire setup acts as a classical microwave filter with a second-order bandpass response. The power handling capability of this filter is examined experimentally using a high-power magnetron source with a frequency of 9.382 GHz, a pulse duration of 1 μs, and a peak power of 25 kW. Based on the results of this experiment, two methods for improving the power handling capability of these multi-layer structures are proposed. In particular, it is demonstrated that emulating the ENG layers with thin perforated metallic sheets with sub-wavelength holes significantly enhances their peak power handling capability. A prototype of such a device is designed, fabricated, and experimentally characterized and it is demonstrated that it can handle extremely high peak power levels. The results presented in this work are expected to be useful in designing microwave filters and frequency selective surfaces that can handle extremely high peak power levels.