Is the ideal of a one-size-fits-all university model--where universities are capable of successfully performing all competencies simultaneously--ever achievable? Has the balance between teaching and research activities grown even more fragile or is such a balance unfeasible? In this paper, we review studies of institutional diversity in higher education and bibliometric analyses. We prose a two-dimensional typology based on diversity in teaching and research by scientific fields. We investigate the diversity in teaching and research, respectively, at the disciplinary level and address whether these primary activities mirror each other. As a measure of mismatch between the two activities, we assess the distribution of students and publications by scientific fields through a clustering analysis. Our study of higher education institutions (HEIs) in Russia reveals that private universities are less diverse both in research and teaching and have high teaching intensity compared to state universities. Expectedly, universities in the rankings have more diverse profiles and they have a higher research intensity.