I suggest you verify that your simulation is generating data according to the assumptions that are used for assessing the coverage probability of the CIs. In the article, "Coverage probability of confidence intervals: A simulation approach," I show an example of simulating and computing CIs. For the example (CIs for a mean), the coverage of the CIs assumes that the data are sampled from a normal distribution. When you simulate data from a normal distribution, you correctly get the 95% coverage probability. However, if you sample from nonnormal data, the coverage probability changes. Depending on the tails of the sampling distribution, you might get higher or lower coverage.
... View more