The energy landscape model of the glass transition has received increasing attention, due to its potential for providing a thermodynamic interpretation of the dynamics of glass-forming liquids. Herein, the idea that the temperature-dependence of the dynamics near the glass temperature ("fragility") can be related to thermodynamic properties is tested for polymers. Previously, for several homologous series, we found deviations from a proposed correlation between fragility and the heat capacity increment at the glass temperature. A survey of 17 polymers likewise indicates no correlation between these two quantities. More recently, the landscape approach to the dynamics of glass-formers was extended to a proposal that fragility can be determined a priori from thermodynamic information. However, for the 11 polymers for which sufficient thermodynamic and relaxation data were available, we find that thermodynamic properties bear no relationship to fragility. Thus, it appears that the status of the glass transition as an important unsolved problem in condensed matter physics remains intact. [S0021-9606(99)51136-7].