Despite the existence of well-defined relationships between cold gas and star formation, there is evidence that some galaxies contain large amounts of H I that do not form stars efficiently. By systematically assessing the link between H I and star formation within a sample of galaxies with extremely high H I masses (log /M > 10), we uncover a population of galaxies with an unexpected combination of high H I masses and low specific star formation rates that exists primarily at stellar masses greater than log M */M ∼ 10.5. We obtained H I maps of 20 galaxies in this population to understand the distribution of the H I and the physical conditions in the galaxies that could be suppressing star formation in the presence of large quantities of H I. We find that all of the galaxies we observed have low H I surface densities in the range in which inefficient star formation is common. The low H I surface densities are likely the main cause of the low specific star formation rates, but there is also some evidence that active galactic nuclei or bulges contribute to the suppression of star formation. The sample's agreement with the global star formation law highlights its usefulness as a tool for understanding galaxies that do not always follow expected relationships. © 2014. The American Astronomical Society. All rights reserved..