Hi dsyfox,
Do you know whether the original PDF was created from a scanned document? And did you convert the file from within Reader or via the ExportPDF website? It sounds like the text in the PDF may have been treated as an image, which can happen if the original PDF was created from a scanned document.
If you'd like me to take a look at the document, let me know and I'll shoot you a private message with my email address.
Best,
Sara