We present a class of interacting nonlocal quantum field theories, in which the CP T invariance is violated while the Lorentz invariance is present. This result rules out a previous claim in the literature that the CP T violation implies the violation of Lorentz invariance. Furthermore, there exists the reciprocal of this theorem, namely that the violation of Lorentz invariance does not lead to the CP T violation, provided that the residual symmetry of Lorentz invariance admits the proper representation theory for the particles. The latter occurs in the case of quantum field theories on a noncommutative space-time, which in place of the broken Lorentz symmetry possesses the twisted Poincaré invariance. With such a CP T -violating interaction and the addition of a C-violating (e.g., electroweak) interaction, the quantum corrections due to the combined interactions could lead to different properties for the particle and antiparticle, including their masses.