I was unable to "fix" this per se. The instructions on this page appeared likely to help, but ultimately didn't, at least in my case.
Except for the one suggestion to revert to an earlier version of my file. Fortunately, through sheer dumb luck, I had saved the file immediately before I pasted in the diagrams. I reverted to that.
I also checked the file I was copying the diagrams from, and it appears that I did something to corrupt the diagrams there as well: I was unable to scale the diagrams even in that original file and got the same error.
It's worth noting that the error had infected other parts of the target file: a disrelated text box produced the same error when I attempted to scale it after the corrupt diagram had been copied into the file.
I reverted to an earlier version of the source file as well, checked that I could scale it in the source file, and then copied the verified-not-corrupt diagram to the target file I'm actually working on.
EDIT: Turns out I spoke too soon. I found that one diagram still gave trouble, even in my earlier version of the source file.
By narrowing down the scope of what I selected (after ungrouping the diagram), I found that there was one SINGLE object, a short line with an arrow on it, that produced the error. Any grouping that it was part of would also produce the error.
I believe that the "contagion" observed earlier was actually due to the text box I was trying to resize being an earlier part of the same text flow that (on a separate page) contained the diagram with this one arrow.
I deleted that arrow and redrew it AND THE PROBLEM CONTINUED.
However, I had several other almost identical arrows in the same diagram that didn't produce the problem. I copied one of these, scaled it a little bit to get it pointing the right way (at an angle), and could then scale the whole diagram successfully.