The settings that are associated with RTF Files are most likely corrupted within your registry.
Try the following.
1) Open up windows explorer
2) Click on Tools, Folder Options, File Types.
3) Scroll down until you find the RTF File type
4) Highlight it and click on the Advanced Button
5) Click on Open, then Click on Edit (If you dont have an open option click on new and call it open)
6) Application used to perform should be set to the following line (It includes all the ")
"C:\Program Files\Windows NT\Accessories\WORDPAD.EXE" "%1"
7) Tick in box for use DDE, Leave DDE Message blank, Application should be WORDPAD, Leave DDE Application not running blank, Topic = System
8.) Click on OK, check the box for confirm open after download. Click on OK again and your RTF files should now be reassociated with wordpad correctly.
Failing that it is possible that something has gone completely awry. I managed to replicate your issue and the above did not really solve it so
try checking if you have a wordpad.inf file in your windows\inf folder. if so then try reinstalling the inf (Right hand click and install).
Then run through the options above and see if it works, it did for me. Although as a final thing I had to change the icon as even though it was showing it when I went to browse for icon it said it could not find and I had to point it at wordpad.exe again.
Hope this helps.