I had a hard time getting a first custom image to load (using Firefox v.29.0.1), but they all eventually loaded in five groups that I own. When I tried to upload even newer images, they were all rejected, with error messages.
I found a workaround that may work with other cover photo upload issues.
- Upload a copy of the image to the Group photo album called "Yahoo! Photo Album" If you don't already have one [I didn't], create one exactly as I just typed it. (You have to have your Photos section activated for this to work.)
- Retry the "Change Cover" button on the cover photo, choosing the same photo that you just uploaded in step #1.
P.S.: I have since been able to change cover photos without first adding them to said album, so this fix may have just cleared some temporary hangup.