ASHINGTON, Dec. 9 — Three years after the Supreme Court reaffirmed its landmark Miranda decision as having "announced a constitutional rule," it was apparent during arguments at the court in two cases on Tuesday that the practical meaning of Miranda and the continued utility of its famous warnings for criminal suspects remain very much in question.

At issue was what use prosecutors may make of evidence obtained after an initial failure by the police to read suspects their Miranda rights before the start of questioning. Each case reflected a common scenario that defense lawyers say will become much more common if the court finds the evidence admissible.

In one case, the result of a federal prosecution in Colorado, the evidence was physical: an illegal gun, the location of which was disclosed to a federal agent by a man who had just been placed under arrest.

When the agent started reciting the Miranda warnings, the suspect, Samuel F. Patane, interrupted by saying that he already knew his rights. In answer to the agent's question, Mr. Patane then said the gun was on a shelf in the bedroom. The United States Court of Appeals for the 10th Circuit, in Denver, ruled that not only could that statement not be introduced, but that the gun itself, as the "fruit" of a Miranda violation, must be kept out of court as well.

In the second case, resulting from a state murder prosecution in Missouri, the police pursued an intentional strategy of questioning a suspect without Miranda warnings, obtaining incriminating admissions before resuming the interrogation after a 20-minute delay. This time, the suspect, Patrice Seibert, was fully advised of her rights, signed a formal waiver form, and incriminated herself again.

The initial statement was clearly inadmissible. The question for the Supreme Court on Tuesday was whether the second statement, following the warnings, should be seen as the product of a fully informed suspect's free will, admissible in court. The Missouri Supreme Court said no, ruling that to admit the second statement in this circumstance would be to encourage deliberate evasions of the Miranda rules.

The Supreme Court ruled in a 1985 case from Oregon that an intervening warning could isolate the initial failure and make a second statement admissible. In that case, however, the initial failure to give the warnings was inadvertent. A central issue in the new case is whether it mattered that the police had deliberately omitted the warnings.

It made no difference, an assistant solicitor general, Irving L. Gornstein, argued for the federal government, which is supporting Missouri's appeal, Missouri v. Seibert, No. 02-1371. "That the initial failure was intentional adds nothing to the psychological force that operates on the suspect, and makes the subsequent warnings no less effective," he said.

Justice Stephen G. Breyer objected. "But what it does do is provide a tremendous incentive for the police to run around Miranda," he said.

Justice Ruth Bader Ginsburg agreed, adding, "Miranda, whatever it has become, has all over it `inform at once.' Now it can be `don't inform until, until you've gotten enough.' "

Arguing for the state, Karen K. Mitchell, Missouri's chief deputy attorney general, said the officers' intention in not providing the warnings in the first instance was irrelevant to the admissibility of the second statement. Only two questions mattered for that purpose, she said: whether the suspect made a "knowing, intelligent and voluntary waiver" of the right to remain silent and to have a lawyer present, and whether the statement was in fact voluntary.

Justice David H. Souter responded that he doubted that the two statements "can really be separated as a matter of simple psychology." He said it was more likely that the second statement reflected not a genuine waiver of rights, but a "throwing up of hands" on the premise that it was too late to take back the earlier self-incrimination. "That's the basic implausibility in your case," Justice Souter told Ms. Mitchell.

But if the government lawyers received skeptical questions from the court's liberal wing, conservative justices were at least as hard on Amy M. Bartholow, an assistant public defender who was urging the court to uphold the Missouri Supreme Court's decision. "I guarantee you that if the court says this practice is O.K., it will become embedded in police procedure," Ms. Bartholow said.

Suppose a second incriminating statement was indeed more likely after the police have obtained an initial statement, Justice Anthony M. Kennedy said to Ms. Bartholow. Then he asked "So what, if it's not coerced?"

Along with Justice Kennedy, Justices Antonin Scalia and Chief Justice William H. Rehnquist observed at several points throughout the two arguments that there was a significant difference between failing to give Miranda warnings and coercing an admission from an unwilling suspect. The former violated the Miranda rule but only the latter actually violated the Fifth Amendment's right against compelled self-incrimination, they said.

This line of discussion raised profound and provocative questions about what the court thought it was doing three years ago when, in Dickerson v. United States, it declared in a 7-to-2 opinion by Chief Justice Rehnquist that Miranda v. Arizona had "announced a constitutional rule" that could not be overturned by Congress.

Regarded as somewhat opaque when it was issued, the Dickerson opinion has become more elusive with time, as shown Tuesday by an exchange between the chief justice and Jill M. Wichlens, an assistant federal public defender from Denver who argued in the Colorado case against the admissibility of the gun.

When Ms. Wichlens referred to the Miranda warnings as "constitutionally required," Chief Justice Rehnquist interrupted her. "The warnings are required in order to make a statement admissible," he said, adding, "It's a conditional thing."

The solicitor general's office, which argued three years ago that Miranda should be preserved, has embraced a carefully confined view of how the procedural protection offered by the Miranda decision relates to the Fifth Amendment. Miranda "provides an extra layer of protection to avoid a violation of the Fifth Amendment right," Michael R. Dreeben, a deputy solicitor general arguing the government's appeal in the Colorado gun case, told the court.

Responding at one point to Mr. Dreeben's assertion of a significant difference between a "compelled" statement and an "unwarned" one, Justice Kennedy said, "You're back into metaphysics."

Mr. Dreeben cheerfully agreed. "It is a little metaphysical," he said.

Under the government's theory of the case, United States v. Patane, No. 02-1183, a piece of physical evidence is "outside the core concern" of the Miranda decision even if derived from a violation of the Miranda rules and from a statement that itself is inadmissible.

Justice Souter objected to Mr. Dreeben, "But there isn't any functional difference in a case like this between admitting the statement and admitting the gun." For the defendant, the protection of Miranda "disappears," he said.

Justice Ginsburg asked: "Are you saying that the police chief can say, `don't worry about his statement, we want the goods?' "

"That is my position," Mr. Dreeben said, adding, "There is a terrible cost to the truth-seeking function of a criminal trial to suppress reliable physical evidence" simply because the statement leading to its discovery was "unwarned."

Justice Souter replied: "The state knows how to avoid paying those costs. It gives the warning."