Once upon a time their was the common law doctrine of the exculpatory no. Basically, if an officer asked you if you robbed the bank and you said "no" you couldn't be prosecuted for the lie because it was a claim of innocence - or maybe just that you were not guilty. This doctrine still exists under Virginia law (although you cannot lie about another person's illegal activities). Under federal law this doctrine has been done away with by statute. We've all seen the cases which result wherein the feds can't make the case against a defendant and then charge him with lying to a federal agent.
Orin asks if that might be unconstitutional.
Admittedly, I haven't researched the history under the common law, but if the doctrine of exculpatory no existed at the time the constitution came into being it strikes me that this could very possibly be a right. Most probably it is a variation upon the defendant's right to not incriminate himself and force the government to prove he is guilty.
On the other hand, if it is just a common law development, in particular one developed after the Bill of Rights was brought into the constitution, it could very well be subject to legislative revision or overruling.
I lean toward the first interpretation because I feel it is truer to the spirit of the constitution.
Anyone out there actually put thought or research into this?