Why can't I open my downloads?
The shortcut "Ctrl+J" takes you to the Downloads Manager, it is basically a journal of downloads. If the file is deleted afterwards it knows nothing about that.
So if you download to a temporary files directly you could have many things clearing that. Likewise if you have Firefox set up to clear your history and that includes downloads you would lose them.
If you are in Private session ("Ctrl+Shift+P") or permanent Private session with Tools > Options > Privacy > "Never Remember History" you would lose them.
Right click on one of the files in the downloads to see where it was downloaded to. Also see about:config settings mentioned in the following along with other problems.