  Dust growth via accretion of gas species has been proposed as the dominant
process to increase the amount of dust in galaxies. We show here that this
hypothesis encounters severe difficulties that make it unfit to explain the
observed UV and IR properties of such systems, particularly at high redshifts.
Dust growth in the diffuse ISM phases is hampered by (a) too slow accretion
rates; (b) too high dust temperatures, and (c) the Coulomb barrier that
effectively blocks accretion. In molecular clouds these problems are largely
alleviated. Grains are cold (but not colder than the CMB temperature, (Tcmb =
20 K at redshift z=6). However, in dense environments accreted materials form
icy water mantles, perhaps with impurities. Mantles are immediately
photo-desorbed as grains return to the diffuse ISM at the end of the cloud
lifetime, thus erasing any memory of the growth. We conclude that dust
attenuating stellar light at high-$z$ must be ready-made stardust largely
produced in supernova ejecta.
