You don't have to apply the CBC mode to complete files. If it is secure for a 1 MB file, I don't see why it would be insecure for 100 parts of a 100 MB file.
If you manage to merge small files into the same blocks, you even gain some privacy because the server can't even tell the number of files anymore.
[1] also has a discussion of the trade-offs of the different modes of operation for whole disk encryption. That seems related here because nobody wants to rewrite the whole disk after changing the first byte.