We present uniaxial (oedometric) compression tests on two cohesive industrially relevant granular materials (cocoa and limestone powder). A comprehensive set of experiments is performed using two devices – the FT4 Powder Rheometer and the custom made lambdameter – in order to investigate the dependence of the powders' behavior on the measurement cell geometries, stress level, relaxation time and applied strain rate. The aspect ratio α, tested with the FT4, is found to play no role for vessels with α ≲ 1 while material characteristics strongly affect the stress–strain response. After compression is stopped, the constant volume stress relaxation is found to follow a power law, consistently for both cohesive powders and for the different testing equipments. A simple (incremental, algebraic) stress evolution model is proposed to describe the relaxation of cohesive powders, which includes a response timescale along with a second, dimensionless relaxation parameter that sets the very small power law, i.e. extremely slow stress relaxation.