Is this happening is just one document or all documents?
There is a newer release (14.1.4) of Office, I don't know if that would have any impact though.
Is Tracked Changes turned on? If it is, turn it off and accept or reject all changes. Save the document, close, reopen and see if drag and drop now works.
In Word > Preferences > Edit > Editing Options what boxes are checked. The default I believe is all checked except the 1st box and 5th. You might also try turning off Smart Cut/Paste as there was problem with it on some systems that used Lion.
Let us know what you find.