  In many models of dark matter (DM), several production mechanisms contribute
to its final abundance, often leading to a non-thermal momentum distribution.
This makes it more difficult to assess whether such a model is consistent with
structure formation observations. We simulate the matter power spectrum for DM
scenarios characterized by at least two temperatures and derive the suppression
of structures at small scales and the expected number of Milky Way dwarf
galaxies from it. This, together with the known bound on the number of
relativistic particle species, $N_\mathrm{eff}$, allows us to obtain
constraints on the parameter space of non-thermally produced DM. We propose a
simple parametrization for non-thermal DM distributions and present a fitting
procedure that can be used to adapt our results to other models.
