When does Rao-Blackwell give you nothing at all?

You already estimate the mean $\mu$ of an i.i.d. sample $X_1, \dots, X_n$ using the sample mean $\bar X$, and $S = \sum_i X_i$ is a sufficient statistic. A colleague suggests running Rao-Blackwell anyway, forming $\mathbb{E}[\bar X \mid S]$, hoping to shave the variance further.

Does Rao-Blackwellizing $\bar X$ reduce its variance? Characterize exactly when the procedure gives no improvement.