  The Galactic Center Excess (GCE) of GeV gamma rays can be explained as a
signal of annihilating dark matter or of emission from unresolved astrophysical
sources, such as millisecond pulsars. Evidence for the latter is provided by a
statistical procedure---referred to as Non-Poissonian Template Fitting
(NPTF)---that distinguishes the smooth distribution of photons expected for
dark matter annihilation from a "clumpy" photon distribution expected for point
sources. In this paper, we perform an extensive study of the NPTF on simulated
data, exploring its ability to recover the flux and luminosity function of
unresolved sources at the Galactic Center. When astrophysical background
emission is perfectly modeled, we find that the NPTF successfully distinguishes
between the dark matter and point source hypotheses when either component makes
up the entirety of the GCE. When the GCE is a mixture of dark matter and point
sources, the NPTF may fail to reconstruct the correct contribution of each
component. We further study the impact of mismodeling the Galactic diffuse
backgrounds, finding that while a dark matter signal could be attributed to
point sources in some outlying cases for the scenarios we consider, the
significance of a true point source signal remains robust. Our work enables us
to comment on a recent study by Leane and Slatyer (2019) that questions prior
NPTF conclusions because the method does not recover an artificial dark matter
signal injected on actual Fermi data. We demonstrate that the failure of the
NPTF to extract an artificial dark matter signal can be natural when point
sources are present in the data---with the effect further exacerbated by the
presence of diffuse mismodeling---and does not on its own invalidate the
conclusions of the NPTF analysis in the Inner Galaxy.
