In many countries entrance to Higher Education is determined by the performance of students in secondary school and/or the scores obtained in national exams. The relative weight of these two scores on the admission decision is a relevant policy topic, given its implication on who is admitted to university. The purpose of this paper is to investigate the relative predictive power of these two measures on the academic performance of students in Higher Education. It makes use of a dataset of Bachelor students from Portuguese Higher Education institutions with detailed information about their characteristics and past achievement results. The measure of academic achievement considered is the Bachelor's final average score. The main finding is that the scores given by teachers in secondary school are better predictors of subsequent performance than the access exam scores. The relevance of factors like working status, social support and gender vary with the reputation of the degree and the type of Higher Education institution, university versus polytechnic. A noteworthy result is that the added predictive contribution of parents' education is always negligible when past success measures are already taken into account.