I've been having the same problem running Word 2008, earlier under Leopard and Snow Leopard, and now, still, under Mountain Lion. It affects only certain documents, and the only common feature that I can identify is that (a) they have a lot of graphics in them, and (b) they're stored in the old .doc format ("compatibility mode"). The solution, for me, has been to open the "save as" menu and save in the .docx format. If you do this, don't ditch the .doc version until you've closely reviewed the .docx one, as the conversion (epecially of graphic elements) is often imperfect. You may need to copy-paste some of the original graphics into the new version.
You can specify the default format for saving documents under Word -> Preferences -> Save. This applies only to new documents; existing documents will be saved in whatever format they are already in.
The reason there's no single, magic fix is that "Unable to Save: Disk Full or Too Many Files Are Open" is a stupid choice of language for what is in fact a generic "unable to save" message. It's triggered by Word not obtaining an acknowledgement that the write to disk was completed successfully. A full disk is just one of many possible causes, so the message is quite misleading.