The talk is about tests and confidence intervals based on a test statistic that has a limit distribution that is discontinuous in a nuisance parameter or the parameter of interest. It is shown that standard fixed critical value (FCV) tests and subsample tests often have asymptotic size - defined as the limit of the finite sample size - that is greater than the nominal level of the test. A precise formula for the asymptotic size of such tests is provided under a general set of high-level conditions that are relatively easy to verify. The asymptotic size is determined by a sequence of parameter values that approach the point of discontinuity of the asymptotic distribution. The problem is not a small sample problem. For every sample size, there can be parameter values for which the test over-rejects the null hypothesis. Analogous results hold for confidence intervals. The talk also covers a hybrid subsample/FCV test that alleviates the problem of over-rejection asymptotically and in some cases eliminates it. In addition, size-corrections to the FCV, subsample, and hybrid tests are discussed that eliminate over-rejection asymptotically. Many examples will be given.\\ Talk time runs until 3:00 PM.