I have been plagued by this problem as well and have been able to solve it! Hopefully this will work for you:
1) Make sure there are no other tabs or open browser windows EXCEPT the one for your editing. I think the system gets confused and doesn't know which window to drop the image into. This solved 99% of my problems (in IE and Firefox) and nothing else that was suggested worked. I originally thought it was a browser problem. This also is related to an earlier post about domain masking. I mask my domains as well and this really confuses the admin. Use the registered URL in the general settings with the /wp-admin/ to get into the admin.
2) If you are using WPMU do not use dashes or mix capital and lowercase letters in your blog name. This might also be true for wordpress, but I use MU. I think this has something to do with the permalink structure that was stated in an earlier post, and I think they are on the right track. But I removed a dash and a capital 'I' in one of my names and images were dropping in again.
I have had this problem since upgrading to 2.6.5 and have been able to solve it for 50+ blogs using the above methodology through 2.7.