The case, Messerschmidt v. Millender, No. 10-704, arose from an early-morning search in 2003 of the home of Augusta Millender. The police were looking for Ms. Millender’s foster son, Jerry Bowen, and for a shotgun he had used in a domestic assault. They found neither. But they did take a gun owned by Ms. Millender, who was then 73 and has since died. The gun was legal, and she said she kept it for self-defense. The police had obtained a warrant to search the home, but lower courts ruled that it violated the Fourth Amendment because parts of it were not supported by probable cause. In particular, they said, the warrant improperly allowed the police to search for “all handguns, shotguns and rifles” and “evidence showing street gang membership.” Chief Justice John G. Roberts Jr., writing for the court, rejected Ms. Millender’s lawsuit. The case did not concern the validity of the warrant, he wrote, though he suggested that there was every reason to think it valid. Rather, he said, the question was whether a lawsuit against the officers was permitted. A 1986 Supreme Court decision, Malley v. Briggs, said police officers should be denied immunity from such lawsuits “only where the warrant application is so lacking in indicia of probable cause as to render official belief in its existence unreasonable.” Chief Justice Roberts, joined in full by five other members of the court, said Ms. Millender’s suit did not fit that exception. Justice Elena Kagan agreed that the search for guns was proper but dissented from the part of the majority opinion concerning gang-related evidence, which she said was not connected to the domestic violence that prompted the search. Justice Sonia Sotomayor, joined by Justice Ruth Bader Ginsburg, issued a heated dissent. In response to the chief justice’s conclusion that the officers’ conduct had been “objectively reasonable,” she wrote, “I could not disagree more.” “It bears repeating that the founders adopted the Fourth Amendment to protect against searches for evidence of unspecified crimes,” Justice Sotomayor wrote. “And merely possessing other firearms is not a crime at all.” She cited District of Columbia v. Heller, the 2008 decision finding a Second Amendment right to keep a gun at home for self-defense. Justice Sotomayor added that she was not persuaded by the fact that the officers’ superiors and a judge had approved the warrant. “Under the majority’s test,” she wrote, “four wrongs apparently make a right.” In a second decision issued Wednesday, the court sidestepped a knotty question in a case over California’s attempt to cut Medicaid payment rates. Speaking from the bench Wednesday morning, Justice Stephen G. Breyer began his summary of the majority opinion with an assessment of his task. “The difficult thing,” he said, “is to explain what this case is about.” In response to the budget crisis in California, state officials there lowered payments to medical providers in the Medicaid program there. A federal law requires states that participate in the program, which provides health care to poor and disabled people, to pay rates “sufficient to enlist enough providers” to ensure that care available under Medicaid is similar to that available to other local residents. There is no question that federal authorities can enforce the law and that states that fail to comply with their obligations face the loss of federal money. The question for the court was whether the providers and Medicaid recipients were themselves entitled to sue over the move. After the court agreed to hear the case, federal authorities concluded that the state law passed muster. The majority returned the case to the lower courts to take account of that development. Chief Justice Roberts, writing for himself and Justices Antonin Scalia, Clarence Thomas and Samuel A. Alito Jr., chastised the majority for failing to address the question of whether private lawsuits in such circumstances are ever proper. The correct answer, he said, was no. The decision concerned three consolidated cases that had been argued on Oct. 3, the first day of the current term, including Douglas v. Independent Living Center of Southern California, No. 09-958. It was the first 5-to-4 ruling of the term, and the justices aligned in one of the two usual configurations in such cases, here with Justice Anthony M. Kennedy joining the court’s four more liberal members. The third decision, PPL Montana v. Montana, No. 10-218, was unanimous. The court, in an opinion by Justice Kennedy, overturned a $41 million judgment by the Montana Supreme Court in favor of Montana and against a power company that owns hydroelectric plants on Montana rivers. The question in the case was whether the portions of the rivers in dispute belonged to the state, and the answer turned on whether they were navigable when Montana joined the union in 1889. Drawing in part on journals kept by Meriwether Lewis during his expedition with William Clark, Justice Kennedy concluded that the state court had used the wrong standard in determining navigability and that at least substantial stretches of one river dotted by waterfalls were plainly not navigable.