This is actually much more commonly useful than the t distribution, in my experience. You can squint at a histogram (or some summary stats), eyeball the stdev, approximate the stderr in your head, and get a pretty good sense of confidence.

I most often find myself doing this for the Bernoulli distribution, where it's also handy to know that the stdev is sqrt(p(1-p)), or "about 1/2 if p is middling, or sqrt(p) when it's small" (and you can flip the polarity to handle p→1).