\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\)

The one-sample t-distribution confidence interval is robust for small or even moderate departures from normality, unless \(n\) is very small. But the prediction and tolerance intervals can be off by quite a lot. Don’t use them for non-normal distributions.