The fastest gait and speed of the largest theropod (carnivorous) dinosaurs, such as Tyrannosaurus, is controversial. Some studies contend that Tyrannosaurus was limited to walking, or at best an 11ms-1 top speed, whereas others argue for at least 20ms-1 running speeds. We demonstrate a method of gauging running ability by estimating the minimum mass of extensor (supportive) muscle needed for fast running. The model's predictions are validated for living alligators and chickens. Applying the method to small dinosaurs corroborates other studies by showing that they could have been competent runners. However, models show that in order to run quickly, an adult Tyrannosaurus would have needed an unreasonably large mass of extensor muscle, even with generous assumptions. Therefore, it is doubtful that Tyrannosaurus and other huge dinosaurs (~6,000kg) were capable runners or could reach high speeds.