I mean, anything is possible, but that seems farfetched to me. The router is typically a hard target for malware unless you have physical or at least LAN access. They are generally pretty locked down and don't execute anything from remote access, they examine packet headers and send them on their way. If it was compromised I'd expect something more nefarious than ruining file transfers too.

The biggest strike against this being a network hardware/driver issue is that normal browsing works. If packets were being screwed up in transit, connections would drop, text and images would be corrupt as well (which the browser would probably choke on). It seems to have an issue only when the disk is involved, when data is being saved.