A B777 Flight Crew received TCAS TA and RA warnings at FL 280. Possible TCAS anomaly.

Narrative

We were cruising at FL 280 after being descended by New York Center. We had no aircraft on our TCAS when we suddenly got a Traffic Alert of an aircraft heading directly at us at +500 feet and approximately 10 miles. The TA quickly became an RA with the immediate instructions to descend accompanied by the visual presentation indicating a descent on my PFD. I disconnected the autopilot and autothrottles and started an immediate descent. At approximately FL 275 we got a clear of traffic advisory and the target completely disappeared from the TCAS. While I was flying the aircraft; my Copilot told NYC that we were descending in compliance with an RA. ATC acted like they did not even know what an RA was and stated that the only aircraft in our area was a Military aircraft at FL250. The Military pilot then volunteered that he had a Mode C failure. We never saw an aircraft above us and were not even looking for aircraft below us that we were in fact descending toward. Apparently we never got closer than 2500 feet of the conflicting (and malfunctioning) aircraft. I climbed back to FL280 after descending only 500'. I can't believe that the military is flying around with such faulty equipment and that ATC does not seem to understand our obligations if we have an RA.