You have admin rights? You could try creating a new user and see if the problem continues.
Are you using anything such as Speed Download? If so, disable it.
Go to preferences and reset where the downloads are supposed to go (I have a folder called downloads). Do a search to make sure they didn't download somewhere else on the computer.
Also, you may have clicked off open files after download, which is a good idea for security purposes. You may have to open the stuffed files.
See if those work.