We consider the problem of assessing the effects of a treatment on duration outcomes using data from a randomized evaluation with noncompliance. For such settings, we derive nonparametric sharp bounds for average and quantile treatment effects addressing three pervasive problems simultaneously: self-selection into the spell of interest, endogenous censoring of the duration outcome, and noncompliance with the assigned treatment. Ignoring any of these issues could yield biased estimates of the effects. Notably, the proposed bounds do not impose the independent censoring assumption - which is commonly used to address censoring but is likely to fail in important settings - or exclusion restrictions to address endogeneity of censoring and selection. Instead, they employ monotonicity and stochastic dominance assumptions. To illustrate the use of these bounds we assess the effects of the Job Corps (JC) training program on its participants' last complete employment spell duration. Our estimated bounds suggest that JC participation may increase the average duration of the last complete employment spell before week 208 after randomization by at least 5.6 log points (5.8 percent) for individuals who comply with their treatment assignment and experience a complete employment spell whether or not they enrolled in JC. The estimated quantile treatment effects suggest the impacts may be heterogeneous, and strengthen our conclusions based on the estimated average effects.