Nested sampling is an increasingly popular technique for Bayesian computation, in particular for multimodal, degenerate problems of moderate to high dimensionality. Without appropriate settings, however, nested sampling software may fail to explore such posteriors correctly; for example producing correlated samples or missing important modes. This paper introduces new diagnostic tests to assess the reliability both of parameter estimation and evidence calculations using nested sampling software, and demonstrates them empirically. We present two new diagnostic plots for nested sampling, and give practical advice for nested sampling software users in astronomy and beyond. Our diagnostic tests and diagrams are implemented in nestcheck: a publicly available Python package for analysing nested sampling calculations, which is compatible with output from MultiNest, PolyChord and dyPolyChord.