Make sure your security software isn't causing this issue. Try to disable security software temporarily to see if that makes a difference.

How big are the files in question?

Go to the Mozilla Add-ons Web Page {web Link} (There’s a lot of good stuff here) and search for a download manager.

This download manager works separately from the browser. The browser tells it what to download, and off it goes.

http://www.freedownloadmanager.org It is a powerful modern download accelerator and organizer for Windows and Mac.