This work highlights and solves problems with the prediction of the compressive strength, limited by local instabilities, of sandwich material compounds based on honeycomb cores and very thin facesheets. Analytical methods in conjunction with periodic finite element unit cell models are utilized for this task. The finite element models are found to be well suited for all kinds of buckling predictions. Different uni- and bi-axial loadings are considered as well as influences of core height, core material, core geometry, and facesheet thickness are investigated. Finally, a new analytical approach is introduced for the treatment of the rather unexpected core cell wall buckling under in-plane compression of the sandwich, which predicts the critical load very accurately.