  By combining experimental constraints from atmospheric and solar neutrino
oscillations and the tritium beta decay endpoint, we infer upper limits of 4.4
eV on the $\nu_\mu$ and $\nu_\tau$ masses, if the universe consists of three
neutrinos. For hierarchical mass spectra $m_3 \gg m_1, m_2$ or $m_3 \simeq m_2
\gg m_1$ we infer that $m_{\nu_\alpha} \lsim 0.08$ eV for $\alpha = e, \mu,
\tau$. In a four neutrino universe, and assuming neutrino oscillations also
account for the LSND experimental results, $m_{\nu_\alpha} \lsim 5.4$ eV. We
also obtain lower limits on the masses.
