If you ask the average citizen, to list all of the Bill of Rights, they probably could not do it, however, it you ask them to list what they could remember, they probably would remember at least a partial bit of the 5th Amendment, and would state that you have the right to not self-incriminate yourself, and further, this might be the sole Amendment, that they would know the number of, probably because they have seen so many criminal trials real or fictionalized on television, to which a defendant has invoked their "5th Amendment right to not testify against himself, under advice of counsel." One thing though a careful reading of the 5th Amendment shows that it states: "… nor shall be compelled in any criminal case to be a witness against himself…" this means that the 5th Amendment does not however stipulate the right to not incriminate yourself in a civil action, because a civil suit is not a criminal suit, and incrimination applies only to criminal cases.
This doesn’t mean, however, that you have to answer all questions in a civil suit, especially if you are concerned that giving answers to certain, specific questions, may present evidence that could be used against you in a criminal suit. This point is extremely valuable because if civil suits can or could be the basis for the later prosecution of criminal suits, than prosecutorial agents would be well within their rights to suggest to the plaintiff to first pursue the civil case, and then have the prosecutorial arm of the State take evidence that would be considered incriminating against the defendant and then use this very testimony given in a civil suit against him in a criminal court of law.
As you might suspect, the foregoing not only seems to be unfair, but would also appear to be illegal or a violation of either civil or criminal codes. It could very well be, depending upon how the legal system looks upon it; what it is for a certainty, is it is not "double jeopardy" because double jeopardy is being subject to the same offense twice, whereas these offenses may have the same root, but are different branches, that is, one is criminal and prosecuted by the State, and the other is civil and litigated by private parties.
In general, but not always, in situations to which there are both civil as well as criminal suits pending, or a tendency for both to come into play, American jurisprudence first pursues the criminal case, and then the civil case, which is the way it should be, as this allows the answers in a civil suit to be made without the fear of criminal prosecution. Additionally, when it isn't clear as to whether there will be a criminal case but it is something that could occur, the defendant is allowed within a civil suit, to not answer specific questions that would be incriminating to him in a criminal trial, till there has been proper resolution of the criminal legal threat or a decision made by a judge compelling such testimony to go forward.
The thing is, judges or the law don't always get it right, and there are cases, famous cases (e.g. Bill Cosby), to which a person has given civil testimony, had the civil case settled or finalized, and then years later, been prosecuted or threatened with prosecution in a criminal case, to which their civil testimony is used against them. This would clearly seem to be a case of being compelled by civil evidence to be a witness against himself, and even a cursory reading of the 5th Amendment, would seemingly make this evidence inadmissible.
Should future litigation follow this pattern, first the civil and then the criminal -- with the prosecution distilling key components of the civil, and thereby melding the formidable State with private parties, this could specifically target individuals and effectively break them.