Monte Carlo tests have exact level when the distribution of the test statistic is free of nuisance parameters. Confidence sets obtained by inverting such tests are also exact but may have a complicated structure. The sets reduce to intervals if the sampling can be organized in a special way. A sufficient condition is that the test statistic is monotone in the parameter of interest when all random drawings are kept fixed. Examples given include models from the one-parameter exponential class. A simple theory quantifying the impact of Monte Carlo uncertainty is also developed.