Say we have a normal distribution with unknown mean and variance and we take $n$ samples from it. Then, the usual confidence interval for the mean uses the $t$-distribution. But if we know that the variance is less than, say, 100, can we use that knowledge to make a smaller confidence interval for the mean?
It should be possible, but the benefit may not be worth the effort. A Wald confidence interval does not account for the sampling variability of the estimated standard error, essentially treating it as known. Inverting this test produces a confidence interval very similar to inverting the t-interval. Any interval you construct that accounts for the sampling variability of the standard error, while knowing that the unknown true population variance is less than 100, will be wider than a Wald interval and shorter than a t-interval. Let me know if I have made any mistakes.