Let $X_i = \theta + \sigma Z_i$, where $Z_i$ are i.i.d. from a distribution $F$, and $-\infty < \theta < \infty$ and $\sigma > 0$ are unknown parameters. If $F$ is $N(0, 1)$, a standard confidence interval for the unknown mean $\theta$ is the $t$-interval $\bar{X} \pm t_{\alpha/2} s/\sqrt n$. The question of conservatism of this interval under nonnormality is considered by evaluating the infimum of its coverage probability when $F$ belongs to a suitably chosen class of distributions $\mathscr{F}$. Some rather surprising phenomena show up. For $\mathscr{F} = \{\text{all symmetric unimodal distributions}\}$ it is found that, for high nominal coverage intervals, the minimum coverage is attained at $U\lbrack -1, 1 \rbrack$ distribution, and the $t$-interval is quite conservative. However, for intervals with low or moderate nominal coverages $(t_{\alpha/2} < 1)$, it is proved that the infimum coverage is zero, thus indicating drastic sensitivity to nonnormality. This phenomenon carries over to more general families of distributions. Our results also relate to robustness of the $P$-value corresponding to the $t$-statistic when the underlying distribution is nonnormal.