  We use Chandra and XMM observations of the globular clusters $\omega$ Cen and
NGC 6397 to measure the spectrum of their quiescent neutron stars (NSs), and
thus to constrain the allowed ranges of mass and radius for each. We also use
Hubble Space Telescope photometry of NGC 6397 to identify a potential optical
companion to the quiescent NS, and find evidence that the companion lacks
hydrogen. We carefully consider a number of systematic problems, and show that
the choices of atmospheric composition, interstellar medium abundances, and
cluster distances can have important effects on the inferred NS mass and
radius. We find that for typical NS masses, the radii of both NSs are
consistent with the 10-13 km range favored by recent nuclear physics
experiments. This removes the evidence suggested by Guillot and collaborators
for an unusually small NS radius, which relied upon the small inferred radius
of the NGC 6397 NS.
