Outside of archaeological digs and the like - if someone was on private property and dug into a Native American burial site, were there laws that stipulated what happened to the bones and artifacts? Or would they be treated as private property? It's a bit morbid to think of people doing that, but I've read about folks that did, and I'm curious if they were acting in accordance with the law or against it.
The vast majority of federal and state public laws pertaining to archaeological site preservation came into law in the 1960s and 1970s. There are two that preceded this period: the Antiquities Act of 1906, and the Historic Sites Act of 1935.
The Antiquities Act protected sites from excavation, destruction and vandalism on public lands. It also authorized the protection through procurement of places of historic, prehistoric and scientific importance. The Historic Sites Act of 1935 was largely a policy statement - that it would be the policy of the of the federal government to protect places of archaeological and historic significance. It gave the National Park Service the responsibility to investigate and research these places. Neither of these laws prohibited excavation or vandalism of burial sites on private property.
There are also State, County and local laws and ordinances that prohibit disturbance of burial sites but I do not have the resources to guide you to all pertinent local rules. Based on my experience in the western US, these rules generally post-date the 1950s and apply to established cemeteries. It is very likely that burial vandalism on private property in the 1930s was legal.
Charles R. McGimsey III 1972 Public Archaeology. Seminar Press.