I have a related cryptosystem that I came up with, but is so obvious I'm sure someone else has invented it first. The idea is to back up a file like so: first, do a rolling-hash based chunking, then encrypt each chunk where the key is the hash of that chunk. Then, upload the chunks to the server, along with a file (encrypted by your personal key) that contains the information needed to decrypt each chunk and reassemble them. If multiple users used this strategy, any files they have in common would result in the same chunks being uploaded. This would let the server provider deduplicate those files (saving space), without giving the server provider the ability to read the files. (Unless they already know exactly which file they're looking for, and just want to test whether you're storing it.)
Tangent: why is it that downloading a large file is such a bad experience on the internet? If you lose internet halfway through, the connection is closed and you're just screwed. I don't think it should be a requirement, but it would be nice if there was some protocol understood by browsers and web servers that would be able to break-up and re-assemble a download request into a prolly tree, so I could pick up downloading where I left off, or only download what changed since the last time I downloaded something.