wget -c -b --content-disposition 'http://example.com/somepath.mp4'
I had some connectivity problems so I had to rerun this command for a few remote files.
According to wget's man page, it will either continue where it left off if possible, or not do anything and leave the existing file alone, but instead it deleted several nearly complete files and started over from the beginning. What the heck?
From the man page: "Beginning with Wget 1.7, if you use -c on a non-empty file, and it turns out that the server does not support continued downloading, Wget will refuse to start the download from scratch, which would effectively ruin existing contents. If you really want the download to start from scratch, remove the file."