The matter power spectrum at comoving scales of (1-40) h^{-1} Mpc is very sensitive to the presence of Warm Dark Matter (WDM) particles with large free streaming lengths. We present constraints on the mass of WDM particles from a combined analysis of the matter power spectrum inferred from the large samples of high resolution high signal-to-noise Lyman-alpha forest data of Kim et al. (2004) and Croft et al. (2002) and the cosmic microwave background data of WMAP. We obtain a lower limit of m_wdm > 550 eV (2 sigma for early decoupled thermal relics and m_wdm > 2.0 keV (2 sigma) for sterile neutrinos. We also investigate the case where in addition to cold dark matter a light thermal gravitino with fixed effective temperature contributes significantly to the matter density. In that case the gravitino density is proportional to its mass, and we find an upper limit m_{3/2} < 16 eV (2 sigma). This translates into a bound on the scale of supersymmetry breaking, Lambda_{susy} < 260 TeV, for models of supersymmetric gauge mediation in which the gravitino is the lightest supersymmetric particle.
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