  Recently several models of traversable wormholes have been proposed which
require only arbitrarily small amounts of negative energy to hold them open
against self-collapse. If the exotic matter is assumed to be provided by
quantum fields, then quantum inequalities can be used to place constraints on
the negative energy densities required. In this paper, we introduce an
alternative method for obtaining constraints on wormhole geometries, using a
recently derived quantum inequality bound on the null-contracted stress-energy
averaged over a timelike worldline. The bound allows us to perform a simplified
analysis of general wormhole models, not just those with small quantities of
exotic matter. We then use it to study, in particular, the models of Visser,
Kar, and Dadhich (VKD) and the models of Kuhfittig. The VKD models are
constrained to be either submicroscopic or to have a large discrepancy between
throat size and curvature radius. A recent model of Kuhfittig is shown to be
non-traversable. This is due to the fact that the throat of his wormhole flares
outward so slowly that light rays and particles, starting from outside the
throat, require an infinite lapse of affine parameter to reach the throat.
