The fact that the Centaurs are the primary source of the Short Period Comets is well established. However, the origin of the Centaurs themselves is still under some debate, with a variety of different source reservoirs being proposed in the last decade. In this work, we suggest that the Neptune Trojans (together with the Jovian Trojans) could represent an additional significant source of Centaurs. Using dynamical simulations of the first Neptune Trojan discovered (2001 QR322), together with integrations following the evolution of clouds of theoretical Neptune Trojans obtained during simulations of planetary migration, we show that the Neptune Trojan population contains a great number of objects which are unstable on both Myr and Gyr timescales. Using individual examples, we show how objects that leave the Neptunian Trojan cloud evolve onto orbits indistinguishable from those of the known Centaurs, before providing a range of estimates of the flux from this region to the Centaur population. With only moderate assumptions, it is shown that the Trojans can contribute a significant proportion of the Centaur population, and may even be the dominant source reservoir. This result is supported by past work on the colours of the Trojans and the Centaurs, but it will take future observations to determine the full scale of the contribution of the escaped Trojans to the Centaur population.