We have the Right to remain silent for a reason. 5th amendment prevents you from self-incrimination. However, what it does NOT do is mean you are able to answer some questions and then not others.

When you invoke your 5th amendment Rights, it is not something that can be done selectively to avoid answering some questions. He gave information to officers and reporters.

But, we live in a post-Constitutional era right now anyhow, so, anything goes.