1. You can do it, and you won’t be held in contempt for failing to testify. Though the provision says that no person “shall be compelled in any criminal case to be a witness against himself,” the Supreme Court has made clear that this extends to compelling a person to testify in a civil case, when that compelled testimony could later be used against him in a criminal case. See, e.g., McCarthy v. Arndstein (1924):
The Government insists, broadly, that the constitutional privilege against self-incrimination does not apply in any civil proceeding. The contrary must be accepted as settled. The privilege is not ordinarily dependent upon the nature of the proceeding in which the testimony is sought or is to be used. It applies alike to civil and criminal proceedings, wherever the answer might tend to subject to criminal responsibility him who gives it.
2. But a decision to take the Fifth may be used against a party in a civil case (if the party is the witness who refuses to testify, or is closely enough connected to the witness). In a criminal case, the judge and the prosecutor may not tell a jury “that it may draw an inference of guilt from a defendant’s failure to testify about facts relevant to his case.” But that’s not so in a civil case, see, e.g., Baxter v. Palmigiano (1976):
[T]he Fifth Amendment does not forbid adverse inferences against parties to civil actions when they refuse to testify in response to probative evidence offered against them.
So if the Center for Medical Progress people refuse to testify, the court may well give this whatever “evidentiary value … [is] warranted by the facts surrounding his case.” [UPDATE: The analysis in this particular item turns out to be correct only as to the federal law and Maryland law parts of the lawsuit, not the California law parts; if the federal law and maybe Maryland law parts get thrown out, a California statute that forbids such comment in civil cases on the refusal to testify would apply. For more on that, see this post.]
3. Finally, a witness who takes the Fifth as to some matters can’t then try to provide testimony on the same subject that is helpful to the side that the witness might want to support. If the Center for Medical Progress people take the Fifth as to what they did, they thus likely wouldn’t be able to testify about facts (if there are such facts) that might help them show that certain conversations weren’t private, that the confidentiality agreements they signed might be invalid, or whatever else they want to prove. And if they do testify about certain matters, their testifying would waive their Fifth Amendment privilege, at least as to those matters. See, e.g., Mitchell v. United States (1999):
It is well established that a witness, in a single proceeding, may not testify voluntarily about a subject and then invoke the privilege against self-incrimination when questioned about the details. The privilege is waived for the matters to which the witness testifies, and the scope of the “waiver is determined by the scope of relevant cross-examination.” …The justifications for the rule of waiver in the testimonial context are evident: A witness may not pick and choose what aspects of a particular subject to discuss without casting doubt on the trustworthiness of the statements and diminishing the integrity of the factual inquiry…. [A] contrary rule “would open the way to distortion of facts by permitting a witness to select any stopping place in the testimony.” It would … “make of the Fifth Amendment not only a humane safeguard against judicially coerced self-disclosure but a positive invitation to mutilate the truth a party offers to tell.”