The effect of compressive stresses on the mechanisms of destruction of thin films and coatings is discussed. The causes of the origin and development of compressive stresses are analyzed. Specific critical values reached by compressive stresses can lead to edge delamination of thin-film structures from the substrate surface and their buckling, or in some instances, it can manifest itself in the form of wrinkling of the coating/substrate system with possible subsequent delamination. The main parameters determining the nature of deformation and failure of thin-film structures are the sign and magnitude of the stresses and the ratio between the film strength and film/substrate interface strength. The size, shape, and growth rate of buckled regions depend on several factors, such as the nature of the stress–strain state and film thickness, the magnitude of compressive stresses, and curvature and interface strength. The regularities of film buckling depend on these parameters in a complex way. They are determined by the concurrence between a decrease in the elastic compression energy of the film and an increase in the surface energy due to the failure of the film/substrate interface. Discussed models can be used in the development of technological recommendations for the application of thin films and coatings.