This is something that varies widely by funding agency- you are correct that NIH grant budget limits are exclusive of indirects, but for the NSF the opposite is true. Different funders handle that differently, and additionally some funders have a cap on the indirect rate that they will allow. Many private foundations will only an indirect rate of 20%, for example. Some corporate grants that I've seen do not allow for any indirects at all.

Depending on your institution, and how they do their accounting, this can have serious negative effects for investigators. At an academic medical center, the majority of PIs are NIH funded, so all of the accounting and finance planning for research assumes a 54% indirect cost rate- so if, for whatever reason, your portfolio of grants doesn't fit that mold, you can have issues. I have known PIs who had multiple very large grants from private foundations, and so were not producing the expected amount of indirects and ended up being a net negative to their department's bottom line. This caused them (and their department chair) all kinds of problems.

Accounting and grant budgeting are two of the things that I wish I'd learned more about in grad school!