Check if any Font Substitution is occurring, by going to File > Options >
Advanced. In the "Show document content" section, click the Font
Substitution button. The resulting dialog will show which fonts are
unavailable and which ones are being substituted for them. Then look
in the C:\Windows\Fonts folder to see whether the "missing" fonts are
really missing.
If the files are in the Fonts folder, maybe they're corrupted or not
properly registered. Running the Office installer in Repair Mode
should fix that.
One way of solving this would be by simply backing up (via right clicking on the values) and deleting all the registry values in HKEY_CURRENT_USER\Software\Microsoft\Office\
Obviously this is an extreme approach that will wipe the preferences, but it appears to do the trick.
If all fails you can try saving the troublesome file in what Word 2010 calls "compatibility mode".