Higher education systems in competitive environments generally present top universities, that are able to attract top scientists, top students and public and private financing, with notable socio-economic benefits in their region. The same does not hold true for non-competitive systems. In this study we will measure the dispersion of research performance within and between universities in the Italian university system, typically non-competitive. We will also investigate the level of correlation that occurs between performance in research and its dispersion in universities. The findings may represent a first benchmark for similar studies in other nations. Furthermore, they lead to policy indications, questioning the effectiveness of selective funding of universities based on national research assessment exercises. The field of observation is composed of all Italian universities active in the hard sciences. Research performance will be evaluated using a bibliometric approach, through publications indexed in the Web of Science between 2004 and 2008.