How small can the light quark masses be?

We derive lower bounds for the combination of light quark masses m_s +m_u and m_d +m_u. The derivation is based on first principles: the analyticity properties of two-point functions of local current operators and the positivity of the corresponding hadronic spectral functions. The bounds follow from the restriction of the sum over all possible hadronic states which can contribute to the hadronic spectral functions to those with the lowest invariant mass. These hadronic contributions are well known phenomenologically, either from experiment or from chiral perturbation theory calculations. The results we find cast serious doubts on the reliability of some of the lattice QCD determinations which have been recently reported.