Nanolaminates such as the Mn + 1AXn (MAX) phases are a material class with ab initio derived elasticity tensors published for over 250 compounds. We have for the first time experimentally determined the full elasticity tensor of the archetype MAX phase, Ti3SiC2, using polycrystalline samples and in situ neutron diffraction. The experimental elastic constants show extreme shear stiffness, with c44 more than five times greater than expected for an isotropic material. Such shear stiffness is quite rare in hexagonal materials and strongly contradicts the predictions of all published MAX phase elastic constants derived from ab initio calculations. It is concluded that second order properties such as elastic moduli derived from ab initio calculations require careful experimental verification. The diffraction technique used currently provides the only method of verification for the elasticity tensor for the majority of new materials where single crystals are not available. © 2010, Institute of Physics