TCASII COMMANDED A DSCNT TOWARDS AN ONCOMING ACFT CAUSING LTSS.

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IN CRUISE FLT AT FL350; CAPT FLYING; WE RECEIVED AN RA ALERT ON TCASII. WE WERE COMMANDED TO DSND WHEREUPON I NOTIFIED ATC AND QUERIED THEM ABOUT THE LOCATION OF ANY CONFLICTING TFC. THEY TOLD US OF TFC AT FL330; BUT NONE AT FL350. WE HAD DSNDED TO ABOUT FL347 AND BECAUSE OF THIS ATC GOT THEIR WARNINGS FOR ACFT WITHIN 5 MI OF EACH OTHER HAVING LESS THAN 2000 FT SEPARATION. WE HAD TO CALL ZOB AND TELL THEM WHY WE HAD TO DEVIATE FROM ASSIGNED ALT. AT THIS TIME WE DID NOT KNOW WHY WE GOT AN RA BUT WE SUSPECT IT TO BE A TCASII GLITCH IN THAT IT WAS READING GHOST TFC. WE GOT SEVERAL MORE SUBSEQUENT TA'S. WE WERE PARALLEL TO OTHER FL350 TFC; 12 MI SEPARATION; SO MAYBE THE TCASII WERE TALKING TO EACH OTHER IN ERROR. WE DID NOT NOTICE HOW LONG THE GHOST WAS DISPLAYED BEFORE GETTING AN RA. WE FELT IT POSSIBLY WAS IN ERROR; BUT ACTED AS IF IT WERE A REAL THREAT. SO; IN SUMMARY; WE DEVIATED FROM ASSIGNED ALT IN REACTING TO A TCASII RA AND IN SO DOING; GOT 300-400 FT LOW; CAUSING AN ALT MIN SEPARATION PROBLEM WITH ANOTHER ACFT. THERE WAS NO CONFLICT WITH THE FL330 TFC; HOWEVER. SUPPLEMENTAL INFO FROM ACN 224875: ATC TOLD US THERE WAS ONLY 35000 FT TFC 20 NM FROM US; NOTHING ANY CLOSER. THERE WAS; HOWEVER; TFC IN FRONT OF US AT 33000 FT AT 4 NM AND WE HAD EXCEEDED THEIR PARAMETERS FOR 4 MORE GHOST TFC CALLOUTS COMING FROM THE SAME DIRECTION ALL AT FL350.