The ternary carbide Ti3SiC2 is the archetype of MAX phases. To date, MAX phases have proven difficult to synthesize as sufficiently large single crystals from which single crystal elastic constants might be obtained. Therefore, the elastic properties not only of Ti3SiC2 but other MAX phases are extensively studied by ab initio methods. Recently single crystal elastic constants were experimentally determined for the first time using neutron diffraction. The experiment revealed extreme shear stiffness which is not only quite rare in hexagonal materials but also strongly contradicts the predictions of all published MAX phase elastic constants from ab initio calculations. In the present paper we would like to show that such shear stiffness can possibly be supported by ab initio calculations and the calculated phonon dispersion along high symmetry directions.