This paper presents the first theoretical and experimental study on the power handling capabilities of electrostatically tunable MEMS cavity filters. The theoretical analysis indicates that the frequency-dependent RF voltage inside a narrowband filter may play an important role in the generation of electromechanical nonlinearities such as frequency response distortion, frequency shift, and bifurcation instability. This analysis also reveals that the filter's power handling capability is dependent on several critical factors, including the capacitive gap, stiffness of the diaphragm actuator, and the overall quality factor (Q) of the evanescent-mode (EVA) resonators. A nonlinear computer-aided design (CAD) model is proposed as a practical tool for capturing the important tradeoffs in high-power design. An EVA tunable resonator and a two-pole 2% filter are fabricated and measured as vehicles to validate the theory and the CAD model. Specifically, a medium-power filter with a tuning range of 2.35-3.21 GHz (1.37:1) and an extracted unloaded quality factor (Qu) of 356-405 shows measured power levels of 23.4 dBm (0.22 W) before bifurcation instability occurs. The measured IIP3 of this filter are 52.1 dBm. The theory and modeling, backed up by the measurements, provide significant insights into the high-power design of electrostatic tunable cavity filters.