The role of intelligence and personality traits in explaining school achievement is crucial. However, their mutual impact on academic success remains unclear due to ambiguous results--the synergistic hypothesis versus the compensatory hypothesis. Additionally, there is a lack of research on representative samples, particularly concerning the distribution of results per intelligence quotient (IQ) scores and among children in the lower grades of primary school. The conducted research aimed to address these gaps. A total of 466 students with an IQ score distribution of M = 103.29 and SD = 15.46 were examined. Multisource assessments, including personality traits, fluid intelligence, and school achievement, were analyzed using Pearson's correlation coefficient and moderation analysis. The results indicated that intellectual level moderated the relationship between extraversion and school achievement in all participants; neuroticism and school achievement in students with IQ < 114; and openness to experience and school grades in students with IQ between 103 and 114, showing a synergistic effect, and in students with intelligence > 107, displaying a compensatory effect. These findings offer some explanation for the two opposing hypotheses regarding the synergistic and compensatory interaction between intelligence and personality traits in explaining school achievements. They also highlight how "cognitive" and "non-cognitive" factors combined may contribute to overall school success.