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We point out that present experimental limits from searches for neutral Higgs bosons at LEP already imply stringent lower bounds on the mass of the charged Higgs boson in the Minimal Supersymmetric Standard Model (MSSM); these bounds are especially severe for low values of \tanb (), where the $H^+ \bar{t} b$ coupling is large. However, these indirect constraints are much weaker in simple extensions of the MSSM Higgs sector involving the introduction of an extra U(1) gauge group or an extra Higgs singlet field; in the latter case charged Higgs bosons can even be light enough to be pair produced at LEP.

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