It is usually assumed that the first-Born approximation for electron-atom ionization becomes valid for the fully differential cross section at sufficiently high impact energies, at least for asymmetric collisions where the projectile suffers only a small energy loss and is scattered by a small angle. Here we investigate this assumption quantitatively for ionization of hydrogen atoms. We find that convergence of the Born approximation to the correct nonrelativistic result is generally achieved only at energies where relativistic effects start to become important. Consequently, the assumption that the Born approximation becomes valid for high energy is inaccurate, since by the time it converges, nonrelativistic scattering theory is not valid.