First thing first
Emergency backup procedure
Drag and drop copy your users file folders (music, pictures, documents etc) to a regular external powered drive as soon as possible.
Do not use, setup or trust only TimeMachine at this troubled time as it will copy corrupted data and we need just your users files safe just in case. TimeMachine cannot be used with a Windows PC.
Unmount and Disconnect all external drives once the backup process is completed.
If you have any over 4GB sized files, those won't copy to a standard formatted FAT32/MSDOS drive from the factory, if you need to save these 4GB+ files use another blank/empty drive and use Disk Utility to format it 1 Partition, Option: GUID, Format: OS X Extended Journaled (or the standard default one for older Mac's).
Mac's cannot copy to nor format a NTFS drive. ExFAT format to be used with a Windows machine must be right click formatted by Windows, not OS X (does it wrong) XP machines can install a free exFAT update from Microsoft. Vista-Win 8 can format exFAT.
Formatting a drive will erase all data on it.
Next target a large folder of items like Movies and (since they are backed up) move them to the Trash and Empty Trash
Now Apple Menu > Restart. The partial downloads should be gone.
Yes, I have already backed up the important stuff a while ago, and have time machine operating on the others...Is there any way to find these without completely wiping things and restarting? Movie files are part of my professional work--I don't see the files in that folder, and I'd rather not take the time to transfer all of those files back and forth if I don't have to...
Your boot drive is full, when it's full it might not boot up.
Thus if you move some rather large files off now as a extra precaution, when you reboot if the partials didn't delete, then your machine will still boot.
1: Use Activity Monitor to take note of the drive space before deleting (backed up) files off the boot drive.
2: Take note of what folder your deleting and Finder > Get Info so you know it's size.
3: Delete the folder (of backed up files) by Trashing and Finder > Empty Trash
4: Take a note of your drive space in Activity Monitor
6: Take note of your drive space in Activity Monitor
IF 4 = 6 then the partial downloads didn't get removed, you machine will still boot though because you deleted more files. If this is so, then we will take it from there.
IF after rebooting you have more free drive space, then you know the hidden files were deleted and you can copy those deleted files back from your backup drive.
OK...will this still be true if I have already rebooted the machine? I had been trying to download the file that didn't work sporadically for a few months before I realized the problem of the invisible files taking up so much space. The entire file size is like 10GB so the partials range from 0-10. If it's just a temporary memory storage that a reboot would repair, wouldn't that have already taken care of things?
ETA: I also deleted all the cookies and site data that were stored from Mega through Chrome.
I think I know what your problem is now.
Apple has caused a lot of confusion with versions, timemachine local backups and other hidden cache files that are causing a lot of headaches with "Other" and being able to get a grip on your true drive space.
Your problems downloading from Mega may have more to do with other factors rather than your amount of free drive space.
If you move a lot of stuff off, freeing up a lot of drive space and then try again you'll likely realize drive space not the real problem.