The 5th Amendment doesn't allow the witness to pick and choose which questions they want to answer.
First, the answer to the question must be realated to some theoretical crime, so, for example, asking for your name is not usually incriminating.
Second, it's more of an area of inquiry analysis, so once you begin testifying about related events, you're probably in too deep.
So basically, the right needs to be invoked as early as possible, and preferably before any testimony is given, though this isn't always realistic, especially in civil depositions and trials. Certain, unexpected lines of inquiry affecting the right can arise. And you need to stfu about anything related (standard for civil law), if not altogether (standard for criminal law), after it's invoked.
I once had a guy refuse to be deposed because he was in the weed business, which is legal in California, but a federal crime. The court upheld his right. Moral of the story, if you want to be virtually testimony proof, get involved in the weed business in California.
He was just a witness though. If you're a civil defendant and invoke the right, the court can and usually will default you and enter judgment in favor of the opposing party. The right to silence re criminal liability is not the right to avoid or secret civil liability.