The United States Court of Appeals for the Second Circuit, which sits in New York, recently answered this question "yes" in the case, Jackler v. Byrne, No 10-5089.

First Amendment protected speech cases for public employees were greatly complicated by the Supreme Court's 2006 decision in a case known as Garcetti v. Ceballos and require close and careful analysis.

You can read more about the Jackson v. Byrne decision at Robert Abell's Kentucky Employment Law Blog.