Language development is not completed when children enter primary school. As the comprehension of connectives (such as although, despite) is important for understanding and producing academic texts and, thus, relevant for school success, we investigated its development and influencing factors across primary school age on the basis of a newly developed and validated test instrument. Using a German sample of 627 students (n = 361 language minority learners) in primary school, results of growth curve models showed students’ initial level of the comprehension of connectives to be negatively related to its growth rate. Additional analyses revealed this association to be mainly due to parental socioeconomic status (SES) rather than students’ language background. In particular, parental SES and students’ receptive grammar impacted initial level as well as growth rate of connective comprehension. Our results point to the necessity of a continuous and early sensitization for the register of academic language especially in the group of students from a low socioeconomic background.