ACR X FALSE TCASII TA RA EVASIVE ACTION TAKEN ON FIRST IGNORED ON SECOND.

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ACR X IN LEVEL CRUISE FLT AT FL350 IN CLR DAYLIGHT. WE WERE TALKING TO ATC ON 132.05. A TCASII TA POPPED UP ON THE DISPLAY AT 12 O'CLOCK AND APPROX 10-12 MI INDICATING LEVEL WITH US AND CLOSING AT A FASTER THAN NORMAL CLOSURE RATE (FOR HEAD-ON TCASII TFC). TA 'TFC' ANNOUNCED. FO ASKED ATC IF THERE WAS TFC AHEAD OF US. AT ABOUT THE SAME TIME; THE TA CHANGED TO RA WITH 'CLB' ANNUNCIATED. I DISCONNECTED THE AUTOPLT WHILE THE FO AND I BOTH LOOKED FOR ANY TFC. AT ABOUT THE SAME TIME THAT ATC WAS TELLING US THEY HAD NO TFC NEAR US; THE RA CHANGED TO 'CLB NOW' WITH THE TARGET NOW ABOUT 12 O'CLOCK; 3-4 MI; LEVEL WITH US AND CLOSING RAPIDLY. I COMMENCED A CLB AND THE TARGET APPEARED TO STAY LEVEL WITH US (NO CLB OR DSND INDICATIONS); THEN DISAPPEARED WHEN NEXT TO OUR TCASII ACFT SYMBOL. ALL TCASII WARNINGS CEASED AT ABOUT THAT POINT. I HAD CLBED APPROX 400 FT AND I RETURNED TO FL350. ATC ONCE AGAIN SAID THERE WAS NO TFC CLOSE TO US. ABOUT 5 MINS LATER TA TFC POPPED UP AT 2 O'CLOCK; LEVEL; AND TRAVELING TOWARD US. ATC ONCE AGAIN SAID THAT THERE WAS NO TFC CLOSE TO US. WE WERE ONCE AGAIN FRANTICALLY LOOKING FOR ANY TFC. WHEN THE TARGET GOT TO ABOUT 5-6 MI FROM US I DISCONNECTED THE AUTOPLT; THE TA BECAME AN RA 'CLB' AND ATC WAS QUERIED ABOUT TFC. AGAIN; ATC SAID NO TFC CLOSE TO US. I OBSERVED THE RA OVERLAPPING A YELLOW TA (BOTH SHOWING LEVEL WITH US) AND DECIDED NOT TO CLB; SINCE NO TFC COULD BE SEEN. RA CHANGED TO 'MONITOR VERT SPD' WHEN THE TARGET WAS WITHIN 2 MI OF US; THEN BOTH TARGETS DISAPPEARED AND ALL WARNINGS CEASED. NO TFC IN EITHER SIT WAS EVER SPOTTED. WE HAD FLOWN THIS PARTICULAR ACFT FOR 2 CONSECUTIVE DAYS AT THE TIME OF THESE OCCURRENCES. THE TCASII SYS HAD TESTED OK; BOTH BEFORE AND AFTER THIS FLT; PLUS ALL KNOWN AND OBSERVED TFC HAD APPEARED TO BE PROPERLY DISPLAYED BY THE TCASII BEFORE AND AFTER THESE OCCURRENCES. MY FIRST DECISION TO FOLLOW THE AR AND CLB WAS BASED ON WHAT SEEMED TO US TO BE RELIABLE TCASII WARNINGS AND INFO. MY SECOND DECISION NOT TO FOLLOW THE RA WAS BASED ON WHAT; AT THAT POINT; APPEARED TO BE SPURIOUS AND INCORRECT TCASII WARNINGS AND INFO. MY CONCERN IS THAT THE TCASII SYS SEEMED TO WORK SO WELL BEFORE AND AFTER THESE 2 OCCURRENCES. IF THERE WAS A PROB WITH THE SYS (TCASII) THEN IT WAS SO INSIDIOUS AS TO CAUSE ME TO LOSE CONFIDENCE IN ITS ABILITY TO PROVIDE RELIABLE AND SAFE INFO. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR STATED THE TCASII HAD WORKED CORRECTLY BEFORE AND AFTER THESE INCIDENTS. RPTR SAID THE COMPANY WAS INVESTIGATING POSSIBLE MIL INTERFERENCE WITH TCASII BY PUTTING FALSE TARGETS IN THE AREA. THIS AREA IS WELL KNOWN FOR SECRET MIL OPS.