  We explore an extended cosmological scenario where the dark matter is an
admixture of cold and additional non-cold species. The mass and temperature of
the non-cold dark matter particles are extracted from a number of cosmological
measurements. Among others, we consider tomographic weak lensing data and Milky
Way dwarf satellite galaxy counts. We also study the potential of these
scenarios in alleviating the existing tensions between local measurements and
Cosmic Microwave Background (CMB) estimates of the $S_8$ parameter, with
$S_8=\sigma_8\sqrt{\Omega_m}$, and of the Hubble constant $H_0$. In principle,
a sub-dominant, non-cold dark matter particle with a mass $m_X\sim$~keV, could
achieve the goals above. However, the preferred ranges for its temperature and
its mass are different when extracted from weak lensing observations and from
Milky Way dwarf satellite galaxy counts, since these two measurements require
suppressions of the matter power spectrum at different scales. Therefore,
solving simultaneously the CMB-weak lensing tensions and the small scale crisis
in the standard cold dark matter picture via only one non-cold dark matter
component seems to be challenging.
