FALCON 2000 HAS MULTIPLE TCAS RA WARNINGS WHILE AT CRUISE.

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WHILE CRUISING AT FL380 ALONG J75 N OF GVE; OUT OF NOWHERE WE GET A TA ON OUR TCAS. IMMEDIATELY THE CTLR (ZDC) INSTRUCTED US TO TURN TO A 270 DEG HDG. YOU COULD TELL BY HIS VOICE HE NEEDED THE TURN IMMEDIATELY. AS WE WERE MAKING A R TURN TO 270 DEGS WE GOT AN RA TELLING US TO CLB. WE FOLLOWED THE RA; AS I REMEMBER; IT WAS COMMANDING A CLB OF AT LEAST 1000 FPM. AS SOON AS WE CLBED THROUGH FL381; THE TCAS ANNOUNCED 'CLR OF CONFLICT.' I TOLD CTR THAT WE RESPONDED TO OUR RA AND THAT OUR TCAS TOLD US WE WERE CLR OF CONFLICT. WE DSNDED BACK TO FL380 AND I TOLD CTR THAT. I ASKED THE CTLR WHERE THAT GUY CAME FROM; AND AFTER A BRIEF PAUSE HE INFORMED ME THAT HE (OTHER ACFT) HAD BEEN GIVEN A TURN AND DSCNT. HE (OTHER ACFT) MADE THE TURN BUT DID NOT DSND TO THE ASSIGNED ALT. AT NO TIME PRIOR TO OR DURING THE TA RA DID WE HAVE VISUAL CONTACT WITH THE OTHER ACFT. SUPPLEMENTAL INFO FROM ACN 710009: WHILE CRUISING AT FL380 ALONG J75 N OF GVE WE RECEIVED A TA ON OUR TCAS. THE CTLR (ZDC) ALERTED US OF THE TFC AT 12 O'CLOCK. OUR TCAS SHOWED HIS ALT AT FL384 AND AS WE LOOKED FOR THE TFC WE BOTH NOTICED THE TFC AT OUR 12:30 - 1 O'CLOCK POS AND WELL BELOW US. OUR TCAS BEGAN TO GIVE US AN RA BUT WE BOTH AGREED THAT THE TFC (THE ONLY TFC ON OUR TCAS) WAS WELL BELOW OUR FL380 CRUISING ALT AND WE REMAINED AT FL380 AND INFORMED THE CTLR THAT THE RPTED TFC WAS WELL BELOW OUR ALT BUT OUR TCAS RPTED HIS ALT TO BE FL384. THE CTLR QUESTIONED THE OTHER ACFT AS TO HIS PRESENT ALT AND WE HEARD HIM RPT TO THE CTLR THAT HE HAD 3 ALTIMETERS RPTING FL370. WE HEARD THE CTLR TELL HIM THAT 2 ACFT HAD RPTED A TA RA WHICH SHOWED HIS ACFT AT FL384. THIS HAPPENED TO BE THE SECOND TA RA WITHIN 20-25 MINS FOR US ALONG J75.