The effect of isotropic pressures P up to 5 kbar on the superconducting transition temperature Tc of the FeSe1−xTex system (x = 0, 0.85, 0.88, 0.90) is studied. For the first time, a change in the sign of the effect of pressure on Tc on going from FeSe to the tellurium-rich alloys is observed. This makes it possible to specify more precisely the form of the dependence of the pressure derivative dTc/dP on composition in this system. This dependence is compared with first principles calculations of the electron structure and magnetism of FeSe, FeTe, and FeSe0.5Te0.5 as functions of pressure, as well as with our earlier experimental data on the effect of pressure on the magnetic susceptibility of the normal state in FeSe and FeTe. This comparison is indicative of a competitive interrelationship between superconductivity and magnetism in tellurium rich FeSe1−xTex compounds.