supply video files to editor in another country
Fedex has the highest bandwidth around... Mailing laptop hard drives, SSDs or USB sticks is probably your best bet and may well be cheaper than upgrading your Internet upload capabilities.

Alternately, if you need a way to handle it over the Internet, I'd actually recommend getting a virtual private server.  You might be able to find a file sharing service that will fit your needs, but often for larger files, it may actually be cheaper to run your own server.  Running your own server also allows for uploading in chunks.

When I need to transfer multiple gigabytes of data over my Internet connection, I use Total Commander (my file manager of choice) to split the large files in to multiple pieces (normally around 100MB each) and then upload them via FTP to my server.  Once they are up there, I re-combine the files in to their original form and run an MD5 verification of the file integrity.  Then, it is easy for someone to quickly download the files.

I've used this for transferring 10 to 20GB of files at a time to people and it took a number of hours to upload even over my 5mbps Internet connection, but if you give it enough time, it is a viable option at lower speed connections as well (as long as your ISP doesn't cap you).  You might want to use smaller file sizes though as my experience is that slower connections are more likely to corrupt a file and smaller file splits means less needs to be uploaded again if there is a failure.