The kinetic and calorimetric fragility indices m of binary As-Se and Se-Te chalcogenide liquids with a wide range of fragility are determined using a combination of parallel plate rheometry, beam bending viscometry, and conventional differential scanning calorimetry (DSC). It is shown that both sets of measurements lead to consistent m values only if the validity of the assumptions often implicit in the methodology for the estimation of m are considered. These assumptions are (i) the glass transition temperature T-g corresponds to a viscosity of similar to 10(12) Pa s and (ii) enthalpy and shear relaxation time scales and /shear are comparable near T-g. Both assumptions are shown to be untenable for highly fragile liquids, for which modulated DSC studies demonstrate that tau(en) >> tau(shear) near T-g. In these cases, the above-mentioned assumptions are shown to lead tau(en) >> tau(shear) consistently higher values for the kinetic fragility compared to its calorimetric counterpart.