Burnout syndrome is a state of physical and mental exhaustion suffered by workers, leading them to become ill. Engagement at Work, meanwhile, is a positive emotional state of energy, enthusiasm and pleasure in relation to the work. This study analyzes the relationship between the constructs of Burnout Syndrome and Engagement at Work through the perceptions of employees of a Public Higher Education Institution. Two scales (MBI-GS and UWES) were used on a sample of 1,511 respondents. The technique adopted was estimation of structural equations with a quantile approach. We observed that the relationship between the constructs has asymmetrical effects, as they presented different impacts along the quantiles. This result reveals an opposite and heterogeneous relationship between Engagement at Work and Burnout Syndrome, indicating that the higher the Engagement at Work, the lower the likelihood of the individual developing burnout. This study enables the managers to understand that in terms of the relationship between the constructs, the expected results, should not be similar throughout the organization. Theoretical and practical contributions can be seen in the areas of organizational behavior and people management.