  [abridged] It has been widely claimed that several lines of observational
evidence point towards a "downsizing" (DS) of the process of galaxy formation
over cosmic time. This behavior is sometimes termed "anti-hierarchical", and
contrasted with the "bottom-up" assembly of the dark matter structures in Cold
Dark Matter models. In this paper we address three different kinds of
observational evidence that have been described as DS: the stellar mass
assembly, star formation rate and the ages of the stellar populations in local
galaxies. We compare a broad compilation of available data-sets with the
predictions of three different semi-analytic models of galaxy formation within
the Lambda-CDM framework. In the data, we see only weak evidence at best of DS
in stellar mass and in star formation rate. We find that, when observational
errors on stellar mass and SFR are taken into account, the models acceptably
reproduce the evolution of massive galaxies, over the entire redshift range
that we consider. However, lower mass galaxies are formed too early in the
models and are too passive at late times. Thus, the models do not correctly
reproduce the DS trend in stellar mass or the archaeological DS, while they
qualitatively reproduce the mass-dependent evolution of the SFR. We demonstrate
that these discrepancies are not solely due to a poor treatment of satellite
galaxies but are mainly connected to the excessively efficient formation of
central galaxies in high-redshift haloes with circular velocities ~100-200
km/s. [abridged]
