Ab initio and density functional methods have been employed to study bridge bonding of aluminum compounds. Results for geometry optimizations and vibrational frequency calculations are found to be consistent with the recent literature. Heats of formation for the aluminum compounds dimethylaluminum hydride and trimethylaluminum are poorly described with density functional theory (DFT) methods including the hybrid DFT method. G2 calculations are closer to experimental values with estimated errors of -1.0 to -2.0 kcal/mol per Al-CH3 bond and -1.9 to -4.1 kcal/mol per Al-H bond. The major finding is that DFT methods poorly represent bridge bonding in aluminum compounds. While ab initio methods (represented by the MP2 method) reproduce experimental values within 2-3 kcal/mol. DFT methods, including the hybrid method, show errors of 5-12 kcal/mol. The DFT methods consistently under-bind the dimers of aluminum compounds with respect to two monomers. Exploration of the hybrid DFT functional shows that a better match between experiment and theory is provided by reducing the contribution of the Becke exchange correction. The binding energies are also found to be sensitive to the choice of correlation functional and the inclusion of "exact exchange". Results for associated units larger than dimers indicate it may be difficult to successfully describe all bridge-bonded aluminum compounds with existing DFT methods.