I submitted a bug report and attached the DOCX file, as directed. I received this response.
"That DOCX file has too many internal elements for the XML library calibre uses to handle. Your best bet is to either split it up and convert the parts individually or alternately you can try exporting as filtered HTML from Word and convert that."
I tried saving it as a filtered HTML, but the resulting formatting sucks. Page breaks are gone, and "scaled characters," such as subscripts and superscripts aren't supported.
What exactly constitutes an "internal element?"
If I split it up and convert the parts individually, as suggested, is there any way to recombine them into a single piece afterwards?