We derive nonparametric sharp bounds on average treatment effects with an instrumental variable (IV) and use them to evaluate the effectiveness of the Job Corps (JC) training program for disadvantaged youth. We concentrate on the population average treatment effect (ATE) and the average treatment effect on the treated (ATT), which are parameters not point identified with an IV under heterogeneous treatment effects. The main assumptions employed to bound the ATE and ATT are monotonicity in the treatment of the average outcomes of specified subpopulations, and mean dominance assumptions across the potential outcomes of these subpopulations. Importantly, the direction of the mean dominance assumptions can be informed from data, and some of our bounds do not require an outcome with bounded support. We employ these bounds to assess the effectiveness of the JC program using data from a randomized social experiment with non-compliance (a common feature of social experiments). Our empirical results indicate that the effect of JC on eligible applicants (the target population) four years after randomization is to increase weekly earnings and employment by at least $24.61 and 4.3 percentage points, respectively, and to decrease yearly dependence on public welfare benefits by at least $84.29. Furthermore, the effect of JC on participants (the treated population) is to increase weekly earnings by between $28.67 and $43.47, increase employment by between 4.9 and 9.3 percentage points, and decrease public benefits received by between $108.72 and $140.29. Our results also point to positive average effects of JC on the labor market outcomes of those individuals who decide not to enroll in JC regardless of their treatment assignment (the so-called never takers), suggesting that these individuals would indeed benefit from participating in JC.