We have examined the atomic theory behind recent constraints on the violation of the Pauli Exclusion Principle derived from experiments that look for X-rays emitted from conductors while a large current is present. We also re-examine the assumptions underlying such experiments. We use the results of these studies to assess pilot measurements to develop an improved test of the Principle. We present an improved limit of \(\frac{1}{2}\beta^{2} < 2.6\times10^{-39}\) on the Pauli Exclusion Principle. This limit is the best to date for interactions between a system of fermions and a fermion that has not previously interacted with that given system. That is, for systems that do not obviously violate the Messiah-Greenberg symmetrization-postulate selection rule.