Answer:
The statement is true. The defendant in a case always has the right to say nothing, but it might make him look guilty.
The right to remain silent arises from the 5th Amendment to the Constitution. The duty of the police to refer to this right goes back to the ruling in Miranda v. Arizona back. However, the trend is creating exceptions. In 1984, the Supreme Court introduced an exception to the Miranda warning, the so-called public safety exception. In 2011, the US Department of Justice ruled that this exception should be used by default in trials against terrorist suspects.
The truth is that for common criminal cases, defendants can remain silent if they wish. However, this can be counterproductive for them, as it can induce the jury to think that the defendant refuses to testify not to be charged, because he has no evidence or alibi to prove his innocence.