Mergers of double neutron stars are considered the most likely progenitors for short gamma-ray bursts. Indeed, such a merger can produce a black hole with a transient accreting torus of nuclear matter, and the conversion of a fraction of the torus mass-energy to radiation can power a gamma-ray burst. Using available binary pulsar observations supported by our extensive evolutionary calculations of double neutron star formation, we demonstrate that the fraction of mergers that can form a black hole-torus system depends very sensitively on the (largely unknown) maximum neutron star mass. We show that the available observations and models put a very stringent constraint on this maximum mass under the assumption that black hole formation is required to produce a short gamma-ray burst in a double neutron star merger. Specifically, we find that the maximum neutron star mass must be within 2–2.5 M. Moreover, a single unambiguous measurement of a neutron star mass above 2.5 M would exclude a black hole-torus central engine model of short gamma-ray bursts in double neutron star mergers. Such an observation would also indicate that if in fact short gamma-ray bursts are connected to neutron star mergers, the gamma-ray burst engine is best explained by the lesser known model invoking a highly magnetized massive neutron star.