The Supreme Court justices, six of whom are grandparents, seemed tempted on Wednesday to rule against grandparents’ interests in the thorny case of Troxel v. Granville, a test of a broad Washington state law that allows any third party to petition for visitation rights in the “best interests” of the child.

Several justices seemed concerned that such a wide-open law would allow anyone to “walk in off the street,” as Justice David Souter put it, and seek visitation even if the child’s parents disapprove. Hypotheticals abounded — an aunt, Justice Sandra Day O’Connor suggested, an accordion player, posited Justice Stephen Breyer — all of whom could intrude on core family privacy, as the justices worried in hourlong oral arguments Wednesday.