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In spite of policymakers’ heightened interest in STEM, evidence suggests that students in Ireland and elsewhere are not reaching their potential in mathematics and science. This study investigated high mathematics and science achievement using Ireland’s data from the Programme for International Student Assessment (PISA). Ireland provides an interesting context because, despite students’ good average mathematics and science performance on national and international assessments, there is a notable absence of high achievers. The results indicated that students’ self-beliefs, learning dispositions, engagement, and socio-economic background significantly predicted high achievement. Discrepancies between the study findings and those on average achievement prompt investigation of student and school characteristics at different performance levels to afford students of varying abilities the opportunity to achieve their potential.