Wage inequality is a highly debated topic in policy and academic circles. Policy makers typically consider that a policy promoting the equalization of education levels among the individuals of a society - pushing everybody towards tertiary education - is a good strategy to fight wage inequality. Academics are more pessimistic. This article stresses that a policy of "tertiary education for all" does not necessarily reduce the overall level of wage inequality. It may reduce wage inequality due to differences in education levels among individuals, but it may also increase wage inequality due to differences in unobserved abilities among individuals.