The notion that gravitation might lead to a breakdown of standard space-time structure at small distances, and that this might affect the propagation of ordinary particles has led to a program to search for violations of Lorentz invariance as a probe of quantum gravity. Initially it was expected that observable macroscopic effects caused by microscopic violations of Lorentz invariance would necessarily be suppressed by at least one power of the small ratio between the Planck length and macroscopic lengths. Here we discuss the implications of the fact that this expectation is in contradiction with standard properties of radiative corrections in quantum field theories. In normal field theories, radiative corrections in the presence of microscopic Lorentz violation give macroscopic Lorentz violation that is suppressed only by the size of Standard Model couplings, in clear conflict with observation. In general, this conclusion can only be avoided by extreme fine tuning of the parameters of the theory.