Many high-profile outbreaks are driven by super-spreading, including HIV, MERS, Ebola, and the SARS-Cov-2 pandemic. That super-spreading is a common feature of epidemics is immutable, however, the relative importance of 2super-spreaders to the outcome of an epidemic, and the individual-level traits that lead to super-spreading, is less clear. For example, an individual may contribute disproportionately to transmission by way of an extremely high contact rate or by way of low recovery, but how these two super-spreaders differ in their effect on epidemiological dynamics is unclear. Furthermore, epidemiological traits may often covary with one another in ways that promote or inhibit super-spreading. What patterns of covariation, and between what traits, are most likely to lead to large epidemics driven by super-spreading? Using stochastic individual-based simulations of an SIR epidemiological model, we explore how variation and covariation between transmission-related traits (contact rate and infectiousness) and duration-related traits (virulence and recovery) of infected individuals affects super-spreading and peak epidemic size. We show that covariation matters when contact rate and infectiousness covary: peak epidemic size is largest when they covary positively and smallest when they covary negatively. We did not see that more super-spreading always leads to larger epidemics, rather, we show that the relationship between super-spreading and peak epidemic size is dependent on which traits are covarying. This suggests that there may not necessarily be any general relationship between the frequency of super-spreading and the size of an epidemic.