Occasionally, maybe once every few weeks, TextEdit refuses to save any documents I currently have open in the application. Any changes I have made to these documents appear in danger of being lost. When this symptom appears, any new documents that I open in TextEdit will similarly refuse to save. It's as if TextEdit decided to go on strike. I've seen this happen with a couple of other applications, but most often it's TextEdit. as well.
Selecting Duplicate doesn't work around the problem, as you can't save the duplicate copy. Similarly, trying to use Save As is also fruitless. It just doesn't work.
Others have reported a similar symptom that appears due to a permissions glitch, but a permissions issue is not the root cause here. Happily, the short-term fix is a simple one. Select "Quit TextEdit" from the File menu. TextEdit should quit immediately, without any prompts to save open documents first.
A brief aside is necessary here. Go to the General system preference and note the option to "Ask to keep changes when closing documents." I keep this option enabled. This forces a prompt to manually save changes when closing.
If you too have this option enabled, don't try closing any individual document windows before quitting the application. If you do, you will get a message asking if you want to save the document. As saving will fail, you are ultimately forced to Cancel and wind up back at square one.
After you successfully quit TextEdit, simply relaunch it. The symptom should now be gone. You will be able to save documents again--until the next time the bug rears its head.
"But what about those unsaved changes?," you may ask. "Are they lost?"
Nope. That's the great thing about the latest versions of OS X. It auto-saves documents. In this case, all your recent edits should be restored when you relaunch TextEdit and open the relevant documents, despite the prior inability to save the files. In fact, assuming you have not selected "Close windows when quitting an application" in that General system preferences, the documents you left open when you quit TextEdit will automatically re-appear when you relaunch--including any new documents that have never been saved at all.
More generally, understanding the implications of these two System Preferences options, and how they interact with the Save, Duplicate, Revert, Close and Quit commands, can get a bit tricky. I recommend spending some time experimenting here, until you are sure things work the way you prefer and expect.
Returning to the "failure to save" glitch, there is a potentially related oddity that sometimes pops up. If you check the Finder icon for a document open in TextEdit, you may find a second document nearby. This file appears to be some sort of cache file for the original document. The file contains the same content as the original file but has a cryptic suffix appended to its name, such as "Bugs & Fixes.rtf.sb-1177f531-DJVlD1." Others have reported this same symptom, which may occur independently of the "failure to save" bug.
I assume these mystery files are related to OS X's auto-save feature and that, when things are working properly, you would not see them. In any case, I can confirm that, after quitting TextEdit, you can safely delete the files. The original documents remain in place with their content intact.