  We show that the present LEPI lower bound on the Standard Model Higgs boson
mass ($M_H\gsim60\GeV$) applies as well to the lightest Higgs boson ($h$) of
the minimal $SU(5)$ and no-scale flipped $SU(5)$ supergravity models. This
result would persist even for the ultimate LEPI lower bound ($M_H\gsim70\GeV$).
We show that this situation is a consequence of a decoupling phenomenon in the
Higgs sector driven by radiative electroweak breaking for increasingly larger
sparticle masses, and thus it should be common to a large class of supergravity
models.
  A consequence of $m_h\gsim60\GeV$ in the minimal $SU(5)$ supergravity model
is the exclusion from the allowed parameter space of `spoiler modes'
($\chi^0_2\to\chi^0_1 h$) which would make the otherwise very promising
trilepton signal in $p\bar p\to\chi^\pm_1\chi^0_2X$ unobservable at Fermilab.
Within this model we also obtain stronger upper bounds on the lighter
neutralino and chargino masses, \ie, $m_{\chi^0_1}\lsim50\GeV$,
$m_{\chi^0_2,\chi^\pm_1}\lsim100\GeV$. This should encourage experimental
searches with existing facilities.
