Negative incremental stiffness is known to occur in structures such as post-buckled flexible tubes and single-cell models. A single foam cell under uniaxial loading buckles and exhibits a non-monotonic S-shaped deformation curve, which is indicative of negative incremental stiffness. Negative stiffness is not observed in bulk materials. For example, individual foam cells display negative stiffness but foams tested in uniaxial compression exhibit a plateau in the stress-strain curve because the buckled cells localize in bands. This behaviour is consistent with the continuum view in which strong ellipticity and, hence, a positive shear modulus G and positive C11 modulus are required for stability, even for a constrained object. It is hypothesized that a solid with negative bulk modulus can be stabilized by control of the surface displacement. Experimentally, foams were hydrostatically compressed by controlled injections of small volumes of water into a plastic chamber, causing volumetric deformation. A negative incremental bulk modulus was observed in a foam with 0.4-mm cell size beyond about 20% volumetric strain. A foam with large cells, 2.5-4 mm in size, was anisotropic and did not exhibit the cell buckling required for negative modulus.