[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]
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