There was, however, one time when I had to deal with an escalated claim. The piece in question was the slow movement of Dvorak's "New World" Symphony. The Symphony has long been in the public domain. However, the slow movement was adapted into the song "Goin' Home", which is still under copyright, and there is a long-standing myth that the song was the inspiration for the Symphony rather than the other way around. Thus when I challenged the copyright claim on my New World video, the claimant came back and insisted their claim (on "Goin' Home") was valid. This notice was of course accompanied by the scary warnings about how if I persisted it could count as a strike against my youtube account.
I called their bluff. I responded with links to every online source I could find showing that Dvorak's work came first and was in the public domain.
I don't know what I would have done had they continued to dig in their heels. Happily, they ended up releasing the claim - days before their claim would have expired anyway - with no elaboration. I'd be curious to hear from anyone else who stuck it through the scary youtube warnings and what the outcome was.
[EDIT: thanks for the clarifications on "own content" and "fair use". I never thought of it that way, but now it seems obvious.]