In the Minimal Supersymmetric Standard Model (MSSM) the existence of an upper bound on the mass of the $CP=+1$ lightest Higgs boson, equal to $m_Z$ at tree--level and $simlt 120 GeV$ after the inclusion of radiative corrections, has important phenomenological consequences for Higgs searches. A similar bound, independent of mass parameters other than the electroweak scale, can be calculated in supersymmetric models with an extended Higgs sector. In models with arbitrary Higgs sectors perturbative up to $10^{16} GeV$, we find, including radiative corrections, $m_h simlt 155 GeV$ for $m_t simlt 190 GeV$.