  Understanding how to optimize electronic band structures for thermoelectrics
is a topic of long-standing interest in the community. Prior models have been
limited to simplified bands and/or scattering models. In this study, we apply
more rigorous scattering treatments to more realistic model band structures -
upward-parabolic bands that inflect to an inverted parabolic behavior -
including cases of multiple bands. In contrast to common descriptors (e.g.,
quality factor and complexity factor), the degree to which multiple pockets
improve thermoelectric performance is bounded by interband scattering and the
relative shapes of the bands. We establish that extremely anisotropic
`flat-and-dispersive' bands, although best-performing in theory, may not
represent a promising design strategy in practice. Critically, we determine
optimum bandwidth, dependent on temperature and lattice thermal conductivity,
from perfect transport cutoffs that can in theory significantly boost $zT$
beyond the values attainable through intrinsic band structures alone. Our
analysis should be widely useful as the thermoelectric research community eyes
$zT>3$.
