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This paper presents an analytical, numerical, and experimental analysis of the breakdown strength of microwave gas-filled RF devices containing sharp corners and wedges. For the idealized case of a wedge-shaped geometry, it is shown that only under certain physical circumstances does the singularity and the concomitant strongly enhanced microwave field determine the breakdown strength. In particular, when diffusion is the dominating loss mechanism for the electron density, breakdown is a volumetric process, and the field singularity does not determine the breakdown threshold. In such situations, excessive accuracy in numerical calculations is not required. Conditions for volumetric and localized breakdown, respectively, are established analytically, and the validity is demonstrated by numerical simulations. Finally, the analysis is extrapolated and compared with experimentally observed breakdown thresholds in commercially available resonators of nonidealized geometry. Good agreement between theoretical predictions and experimental results is demonstrated.