  For non-universal gaugino masses, collider experiments do not provide any
lower bound on the mass of the lightest neutralino. We review the
supersymmetric parameter space which leads to light neutralinos, $M_\lsp \lsim
{\cal O}(1\gev)$, and find that such neutralinos are almost pure bino. In light
of this, we examine the neutralino lower mass bound obtained from supernova
1987A (SN1987A). We consider the production of binos in both electron-positron
annihilation and nucleon-nucleon binostrahlung. For electron-positron
annihilation, we take into account the radial and temporal dependence of the
temperature and degeneracy of the supernova core. We also separately consider
the Raffelt criterion and show that the two lead to consistent results. For the
case of bino production in $NN$ collisions, we use the Raffelt criterion and
incorporate recent advances in the understanding of the strong-interaction part
of the calculation in order to estimate the impact of bino radiation on the
SN1987A neutrino signal. Considering these two bino production channels allows
us to determine separate and combined limits on the neutralino mass as a
function of the selectron and squark masses. For $M_\lsp \sim 100 \mev$ values
of the selectron mass between 300 and 900 GeV are inconsistent with the
supernova neutrino signal. On the other hand, in contrast to previous works, we
find that SN1987A provides almost no bound on the squark masses: only a small
window of values around 300 GeV can be excluded, and even then this window
closes once $M_\lsp \gsim 20 \mev$.
