Photinos or higgsinos with mass O(100) eV are not excluded by cosmological considerations, and their radiative decays could be responsible for the surprisingly large ultra-violet background recently detected at a red-shift z∼4. The agreement of the neutrino data from supernova 1987A with standard expectations severely restricts the energy which could have been emitted via such light photinos or higgsinos, and hence constrains the parameters of models in which they appear. In the low-mass photino case, we find that squark masses between ∼60 GeV and ∼2.5 TeV are excluded. This together with laboratory limits excludes the range of squark masses generally favoured by naturalness arguments. In the low-mass higgsino case, we exclude much of the range of ratios of Higgs VEVs favoured by many models.