  We consider theoretically effects of random charged impurity disorder on the
{\it quality} of high-mobility two dimensional (2D) semiconductor structures,
explicitly demonstrating that the sample mobility is not necessarily a reliable
or universal indicator of the sample quality in high-mobility modulation-doped
2D GaAs structures because, depending on the specific system property of
interest, mobility and quality may be controlled by different aspects of the
underlying disorder distribution, particularly since these systems are
dominated by long-range Coulomb disorder from both near and far random quenched
charged impurities. We show that in the presence of both channel and remote
charged impurity scattering, which is a generic situation in modulation-doped
high-mobility 2D carrier systems, it is quite possible for higher (lower)
mobility structures to have lower (higher) quality as measured by the
disorder-induced single-particle level broadening. In particular, we establish
that there is no reason to expect a unique relationship between mobility and
quality in 2D semiconductor structures as both are independent functionals of
the disorder distribution, and are therefore, in principle, independent of each
other. Using a simple, but reasonably realistic, ``2-impurity'' minimal model
of the disorder distribution, we provide concrete examples of situations where
higher (lower) mobilities correspond to lower (higher) sample qualities. We
discuss experimental implications of our theoretical results and comment on
possible strategies for future improvement of 2D sample quality.
