  The time of passage of the transmitted wave packet in a tunneling collision
of a quantum particle with a square potential barrier becomes independent of
the barrier width in a range of barrier thickness. This is the Hartman effect,
which has been frequently associated with ``superluminality''. A fundamental
limitation on the effect is set by non-relativistic ``causality conditions''.
We demonstrate first that the causality conditions impose more restrictive
bounds on the negative time delays (time advancements) when no bound states are
present. These restrictive bounds are in agreement with a naive, and generally
false, causality argument based on the positivity of the ``extrapolated phase
time'', one of the quantities proposed to characterize the duration of the
barrier's traversal. Nevertheless, square wells may in fact lead to much larger
advancements than square barriers. We point out that close to thresholds of new
bound states the time advancement increases considerably, while, at the same
time, the transmission probability is large, which facilitates the possible
observation of the enhanced time advancement.
