I have an iphone 5 that I cannot backup. I have tried to backup via icloud as well as via computer backup but neither works. When I try to backup to my computer I get an error that says iTunes could not back up the iPhone because the iPhone disconnected. I am using a brand new OEM Apple cable. When I attempt to backup via icloud I get an error that just says "the last backup could not be completed."
Check the lighting port on your iPhone 5, make sure there's no signs of corrosion/broken pins/debris/accumulated lint or dust. When your cable is connected it should sit flushed with no pivoting space, if it doesn't chancing are there are lint or dust stuck in there, carefully remove them using a small tool.
Are you using iOS 8 beta 5 by any chance? If so, that might be the cause of the local backup issue; see here: http://forums.macrumors.com/showthread.php?t=1766005&highlight=backup
iCloud backups can timeout if the wifi is too slow and it's backing up a large amount of data. Consider revising which Apps get backed up to iCloud, and maybe disable (at least temporarily) backing up photos.
Check also your iCloud backup usage. Make sure there's enough room!
Settings > General > Usage > iCloud.