A person in California who explored this web site pointed out that in my discussion of this question, I had omitted California's Lybarger v. City of Los Angeles (40 Cal.3d 822, 1985). In this case, the state's Supreme Court found that when a public employee might be charged with a criminal offense, management must advise them of their rights. The Court based this ruling on California's Public Safety Officers Procedural Bill of Rights Act, which states that once it is "deemed" that an employee might be charged with a criminal offense, the employee must be "immediately informed of his constitutional rights."
In this case, the employee (Lybarger) was a Los Angeles police officer whose unit was under investigation for a variety of potentially criminal offenses. When questioned by IAD, he was told that a criminal offense was pending, and that if he refused to answer questions, he would be considered insubordinate and could lose his job. He still refused to answer questions, and was terminated.
The state Supreme Court found that "the discipline . . . must be set aside because appellant was never advised that any statements he made could not be used against him in a subsequent criminal proceeding" (825).
It is important for readers to note that this decision only applies in California. However, similar requirements do apply in federal service and in the states covered by the United States Court of Appeals for the Seventh Circuit (Illinois, Indiana, Wisconsin), and possibly also the states of the Second Circuit (Connecticut, New York, Vermont).
In regard to Lybarger, a couple additional points are worth making.
First, this case illustrates the benefit of state legislation clarifying and codifying Garrity Rights in the form of a "law enforcement officers' bill of rights." These statutes can streamline understanding of Garrity, and, as we can see in Lybarger, they can create tangible rights over and above those derived from the Garrity-related case law. Without the state legislation, there might have been no clear obligation for management to advise Lybarger of his rights.
Other states have enacted similar legislation. While I applaud the creation of such "law enforcement officers' bills of rights," it does lead me to the question: what about other public employees in that state? This excellent legislation goes beyond Garrity and provides additional protections - but only for law enforcement employees. Teachers, plow truck drivers, social workers, etc. are left with a lesser set of rights. While I applaud the existence of a "law enforcement officers' bill of rights," why not a "public employees' bill of rights"?
Second, in this case, I would argue that Lybarger would possibly have been protected by Garrity regardless of the legislation requiring that he be advised of his rights. He'd been brought in for questioning relative to possible criminal offenses, and had been told that if he refused to answer questions, he could be terminated. It seems to me that at that point, his answers are compelled and are therefore protected, regardless of any advisement of his rights. See Uniformed Sanitation I.
However, I say this only as a musing. The employee in this case had much better protection due to the existence of the Public Safety Officers Procedural Bill of Rights Act.