Scientific output is not a linear function of amounts of federal grant support to individual investigators. As funding per investigator increases beyond a certain point, productivity decreases. This study reports that such diminishing marginal returns also apply for National Institutes of Health (NIH) research project grant funding to institutions. Analyses of data (2006-2015) for a representative cross-section of institutions, whose amounts of funding ranged from $3 million to $440 million per year, revealed robust inverse correlations between funding (per institution, per award, per investigator) and scientific output (publication productivity and citation impact productivity). Interestingly, prestigious institutions had on average 65% higher grant application success rates and 50% larger award sizes, whereas less-prestigious institutions produced 65% more publications and had a 35% higher citation impact per dollar of funding. These findings suggest that implicit biases and social prestige mechanisms (e.g., the Matthew effect) have a powerful impact on where NIH grant dollars go and the net return on taxpayers investments. They support evidence-based changes in funding policy geared towards a more equitable, more diverse and more productive distribution of federal support for scientific research. Success rate/productivity metrics developed for this study provide an impartial, empirically based mechanism to do so.