You are in custody, are read your Miranda rights, and want to invoke your right to remain silent. So you remain silent. Have you actually invoked the privilege against self-incrimination, or can police officers continue to question you?
The U.S. Supreme Court considered this situation in a 2010 decision, Berghuis v. Thompkins. (560 U.S. 370 (2010).) In that case, two officers provided the suspect with the Miranda warnings and interrogated him about a fatal shooting. During the three-hour interrogation, the suspect sat mostly silent. He never said that he wanted to remain silent, didn't want to talk to the police, or wanted a lawyer. Toward the end of the interview, the officers asked him, "Do you pray to God to forgive you for shooting that boy down?" The suspect answered, "Yes." He later moved to suppress his statements during the interrogation.
The Court held that the defendant didn't invoke his right to remain silent by being silent during the interrogation. It said that, to invoke either that right or the right to counsel, a suspect must do so unambiguously. It also said that the police don't have to end an interrogation or ask questions to clarify whether the suspect wants to invoke the Miranda rights if the suspect says nothing or makes an "ambiguous or equivocal" statement. (For related reading, see Miranda: Claiming the Right to Counsel.)
To the Supreme Court, the suspect would have enforced his right to end police questioning by saying that he wanted to remain silent or didn't want to talk. But the Court determined that he waived his right to remain silent. The facts suggested that he understood the Miranda warning, and that he nevertheless chose to speak—eventually. To the Court, the response to the question about praying for forgiveness showed a "course of conduct indicating waiver of the right to remain silent." (The Court said that the suspect's "sporadic answers" throughout the interview before the answer to the prayer question supported this conclusion.)
So, under Berghuis v. Thompkins, the police may continue to interrogate a suspect who has received the Miranda warning and hasn't clearly invoked her rights—even if she hasn't yet waived them.
(To learn about police encounters where the suspect isn't under arrest, see Talking to the Police When You're Not in Custody.)
The Supreme Court sets the standard when it comes to the federal Constitution, but states are free to interpret their own constitutions to provide defendants more rights. So, the law may vary somewhat depending on where your case is. For that reason, among many others, it's wise to consult a knowledgeable criminal defense lawyer if you've been arrested.
For further information on officers persisting with questions, including the difference between the suspect invoking the right to silence and the right to counsel, see Police Questioning After the Suspect Claims Miranda.