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 Tg corresponds to a viscosity of ∼1012 Pa s and (ii) enthalpy and shear relaxation time scales τen and τshear are comparable near Tg. Both assumptions are shown to be untenable for highly fragile liquids, for which modulated DSC studies demonstrate that τen ≫ τshear near Tg. In these cases, the above-mentioned assumptions are shown to lead to consistently higher values for the kinetic fragility compared to its calorimetric counterpart.