Three genera of very large volant birds existed for most of the Pliocene: the Pelagornithidae seabirds; the large North American Teratornithidae and the stork Leptoptilos falconeri in Africa and Asia. All became extinct around 3 Ma. The reasons for their demise are puzzling, as the Pelagornithidae had a world-wide evolutionary history of more than 50 Ma, smaller teratorns were still extant in the Holocene and smaller stork species are still globally extant. Extant large birds have a common critical takeoff airspeed suggesting a biomechanical limit in terms of power, risk and launch speed, and simulations of the flight of these extinct species suggest that at 1 bar they would have exceeded this value. Estimates for the Late Pliocene atmospheric density are derived from marine and terrestrial isotopes as well as resin chemistry, both approaches suggesting a value of about 1.2 bar, which drops to present levels during the period 3.3 to 2.6 Ma, thus a loss in atmospheric density may have caused biomechanical and ecological stress contributing to their extinction and/or development of smaller forms. This hypothesis is examined in terms of a possible mechanism of atmospheric mass loss and how this would be seen in the geological record. At 1.2 bar all the extinct species present takeoff airspeeds similar to large extant volant birds and which match the expected power and kinetic energy levels.