You'll get duplicates because there's no way for Time Machine to know if the files are moved or copies - a good thing if you happen to have files in various locations with no connection whatsoever aside from having the same name.
Originally Posted by Bear
That's why it's good to have a Time Machine drive that's larger than the disk being backed up.
If you don't though (or the drive isn't that much larger), you can go into Time Machine preferences to prevent the backup of specific folders. Also, enter Time machine and delete all of the backups of the old-now-duplicate files.