  The generic bound of quantum speed limit time (the minimal evolution time)
for a qubit system interacting with structural environment is investigated. We
define a new bound for the quantum speed limit. It is shown that the
non-Markovianity and the population of the excited state can fail to signal the
quantum evolution acceleration, but the initial-state dependence is an
important factor. In particular, we find that different quantum speed limits
could produce contradictory predictions on the quantum evolution acceleration.
