We present an extensive analysis of the proposed relationship [T. Scopigno et al., Science 302, 849 (2003)] between the fragility of glass-forming liquids and the nonergodicity factor as measured by inelastic x-ray scattering. We test the robustness of the correlation through the investigation of the relative change under pressure of the speed of sound, nonergodicity factor, and broadening of the acoustic exitations of a molecular glass former, cumene, and of a polymer, polyisobutylene. For polyisobutylene, we also perform a similar study by varying its molecular weight. Moreover, we have included new results on liquids presenting an exceptionally high fragility index m under ambient conditions. We show that the linear relation, proposed by Scopigno et al. [Science 302, 849 (2003)] between fragility, measured in the liquid state, and the slope alpha of the inverse nonergodicity factor as a function of T/T(g), measured in the glassy state, is not verified when increasing the data base. In particular, while there is still a trend in the suggested direction at atmospheric pressure, its consistency is not maintained by introducing pressure as an extra control parameter modifying the fragility: whatever is the variation in the isobaric fragility, the inverse nonergodicity factor increases or remains constant within the error bars, and one observes a systematic increase in the slope alpha when the temperature is scaled by T(g)(P). To avoid any particular aspects that might cause the relation to fail, we have replaced the fragility by other related properties often evoked, e.g., thermodynamic fragility, for the understanding of its concept. Moreover, we find, as previously proposed by two of us [K. Niss and C. Alba-Simionesco, Phys. Rev. B 74, 024205 (2006)], that the nonergodicity factor evaluated at the glass transition qualitatively reflects the effect of density on the relaxation time even though in this case no clear quantitative correlations appear.