Sounds like you have file residue in your hard drive, this occurs when over the course of time, files are saved and deleted on your hard drive causing a build up of useless small pieces of files which haven't been erased properly. These pieces can lead to problems with file functionality.
See what you need to do is go down to the beach and get you some sand. Remove the hard drive and rub some sand on it. Next, pour a spoonfull of sand into the fan at the back of the 360. Replace the hard drive and power on your console, allow the fan to expel the sand from inside the 360. Your AVI files should now play as normal. Hope this helps.