We independently determine the bulk and shear moduli of compressed microgel suspensions and the bulk modulus of individual microgel particles and find that the elastic behavior of the suspension reflects the degree of compression of the particles. This feature, which is distinct from other soft materials such as emulsions or foams, can give rise to an unusually large difference between the bulk and shear moduli of the suspension. Our results extend our understanding of soft materials to systems based on compressible objects, opening up possibilities for engineering materials with drastically different responses to shear and compression.