From the linked article:

Clark County property records show Cliven Bundy’s parents moved from Bundyville, Arizona and bought the 160 acre ranch in 1948 from Raoul and Ruth Leavitt.

Water rights were transferred too, but only to the ranch, not the federally managed land surrounding it. Court records show Bundy family cattle didn’t start grazing on that land until 1954.

Also some research on the history of the Bundy family

If just hanging around in the area gives perpetual grazing rights, I imagine the Native Americans have a better claim.