A witness takes the Fifth. Can the prosecutor and the court inquire into the grounds? No.
Can the witness be punished for not answering questions? Only if he has been ordered to answer.
When can the court order a witness to answer a question? Only if it is ‘perfectly clear, from a careful consideration of all the circumstances in the case that the answer cannot possibly’ tend to incriminate the witness. In those circumstances, even if the judge is wrong about the incriminatory nature of the testimony, an order to answer is a grant of immunity (because ordered testimony is compelled, and compelled testimony is inadmissible unless the witness has explicitly waived his Fifth-Amendment rights).
The prosecutor can always give use immunity and thereby compel a witness to testify. After a witness has been given immunity and answered questions, can he be punished because his answers are not inculpatory? No.

So as a practical matter (which is after all what really matters when a witness is deciding to take the Fifth) any witness can seek the protection of the Fifth Amendment and force the prosecution (or in some circuits the court) to immunize him.

Apropos of Jimbo’s question, in some circuits federal courts may order judicial use immunity if a witness’s testimony is material, exculpatory, and otherwise unavailable. Interesting. In the Fifth Circuit, even, the court may force the government to immunize defense witnesses if the government has abused its immunity power.

“Can”? Absolutely.

You are representing Defendant X on Charge Y. You believe X to be innocent because of a credible alibi witness Z, the ex-girlfriend of X. Z, if she testifies truthfully, will say that X and her were in the throes of passsion all night long when the crime is alleged to have taken place.

For whatever reason, once called to testify by you, Z refuses to testify as to X’s whereabout on the night in question. Maybe there’s been a falling out between X and Z, maybe Z doesn’t want her husband to know what was going down. Whatever the reason, Z refuses to testify and “takes the Fifth.”

Z was not engaged in any illegal behavior. Without the testimony of Z, your client will probably be convicted and spend the rest of his life in prison.

Should the Fifth Amendment, the protection against incriminating one’s self, be able to be twisted in such a manner?

Taking the Fifth is never punishable, even when in the prosecutor’s opinion it works to obstruct justice.

Telling the truth, however, is often punishable when the prosecutor thinks the truth is otherwise.

The only legal way a prosecutor can compel someone to talk over an invocation of the Fifth is to give him immunity. A court order to answer a question, made over an invocation of the Fifth, acts as a grant of use immunity.

The person who thinks he is innocent is almost universally better off not talking without first receiving immunity because he has no idea what kind of liability talking might create.

No, they don’t. Prosecutors and the courts have the authority to subpoena and compel people to testify. Failure to testify is punishable by contempt.

The right not to testify only applied to incriminating statements. Truly innocent people do not make incriminating statement because they are innocent.

There is no right in the Constitution to obstruct justice.

Even if the witness is telling you the truth, this is an excellent example of someone who’s entitled to take the Fifth unless immunized. The right to take the Fifth doesn’t depend on the person having a “good” reason or, indeed, any reason at all.

It doesn’t get him out of court, though: the prosecutor’s correct response would be, “that’s fine; we’ll deal with the Fifth Amendment when you come to court on Tuesday, but meanwhile you’re still under subpoena, and if you’re not there I’ll ask the judge to send a deputy out to your work and bring you in to court.”