You also have to vocally state that you are invoking your right to remain silent. Just remaining silent is apparently not enough.

Additionally, the request for a lawyer must be unequivocal. Not "I think I need a lawyer," as much as any reasonable person would consider that as a request for a lawyer. McDaniel, the guy in the linked case got railroaded after saying that he thinks he would rather have a lawyer there to speak for him, and the claim that the questioning should have stopped was dismissed because he hadn't requested a lawyer, only that he thought he needed a lawyer.

Judges bend over backwards to let police mess with our rights, so clarity and assertiveness are a must.