There are conflicting theoretical assumptions about the development of general cognitive abilities in childhood: On the one hand, a higher initial level of abilities has been suggested to facilitate ability improvement, for example, prior knowledge fosters the acquisition of new knowledge (Matthew effect). On the other hand, it has been argued that school education with its special focus on promoting less able students results in a compensation effect. A third hypothesis is that the development of cognitive abilities is—as an outcome of the opposing effects—overall independent of the initial state. In this study, 1,102 elementary students in 3rd Grade worked on two versions of the Berlin Test of Fluid and Crystallized Intelligence at two time points with an interval of five months. Beside the question of how initial state and growth are related (Matthew vs. compensation effect), we considered performance gains in fluid intelligence (gf) and crystallized intelligence (gc) as well as cross-lagged effects in a bivariate latent change score model. Both for gf and gc there was a strong compensation effect. Mean change was more pronounced in gf than in gc. We considered student characteristics (interest and self-concept), family background (socio-economic status, parental education) and classroom characteristics (teaching styles) in a series of prediction models to explain these changes in gf and gc. Although several predictors were included, only few had a significant contribution. Several methodological and content-related reasons are discussed to account for the unexpectedly negligible effects found for most of the covariates.