Density functional calculations are reported for the g and A tensors of TiF3. The calculated magnetic parameters are in good (g and A(dip)) or reasonable (A(isotropic)) agreement with experiment. This lends credibility to the calculated electronic structure, which however differs markedly from the one deduced originally from the ESR data in that the unpaired electron is predominantly 3d rather than 4s. The approximations that enter the conventional qualitative deductions from ESR data are analyzed and verified by explicit calculations. This highlights the pitfalls of the qualitative treatment. It is traced in detail how the present electronic structure leads to calculated ESR parameters that agree well with experiment and how the usual approximations of the qualitative treatment lead one to infer a different electronic structure.