We have used accurate ab initio quantum chemistry calculations together with a simple model to study the hydrogen storage capacity of metal-organic and covalent-organic frameworks by spillover. Recent experiments by Tsao et al. [J. Am. Chem. Soc. 131, 1404 (2009)] {based on an earlier work by Li and Yang [J. Am. Chem. Soc. 128, 8136 (2006)]} have found that IRMOF-8 with bridged Pt catalysts can reversibly store up to 4.7 wt % of hydrogen at room temperature and 100 bar. We have calculated the binding energy for multiple H atoms on model molecules. By counting active storage sites, we predict a saturation excess storage density at room temperature of 5.0 wt % for IRMOF-8. We also predict storage densities of 4.5 wt % for IRMOF-1, 5.4 wt % for MOF-177, 4.5 wt % for COF-1, and 5.7 wt % for IRMOF-15 and IRMOF-16. This suggests that the current experimental H storage results for IRMOF-8 are well optimized. However, for other materials such as MOF-177 and COF-1, the experimental results are not yet optimized, and significantly more H can be stored on these materials. We also find that significant strain will result from shrinkage of the linker molecules as H atoms are loaded onto the crystals.