  The angular momentum of any quantum system should be {\it unambiguously}
quantized. We show that such a quantization fails for a pure Dirac monopole due
to a previously overlooked field angular momentum from the monopole-electric
charge system coming from the magnetic field of the Dirac string and the
electric field of the charge. Applying the point-splitting method to the
monopole-charge system yields a total angular momentum which obeys the standard
angular momentum algebra, but which is gauge {\it variant}. In contrast it is
possible to properly quantize the angular momentum of a topological 't
Hooft-Polyakov monopole plus charge. This implies that pure Dirac monopoles are
not viable -- only 't Hooft-Polyakov monopoles are theoretically consistent
with angular momentum quantization and gauge invariance.
