Would you give examples of rights claimed to public domain material?
OK, this is a question where personal opinion comes in, since one is really talking about an arrangement.

An artist records an old song that is in public domain, arranging it to suit himself. He and the recording company are entitled to copyright the arrangement and to benefit from sales.
This does not stop another artist from making his own arrangement of the old song, and obtaining copyright on his version. An argument may develop if the first with copyright judges that the newcomer is too close to his own in his version.

I have been playing "Songs of the Cowboys," public domain material from Jack Thorp's books but arranged and sung by Mark Gardner and Rex Rideout. The cd is clearly marked Copyright 2005 Palace of the Governors, Museum of New Mexico (copyright holders).
Do you dispute their right to copyright?