A B767-200 CLBING WBOUND OUT OF JFK CAME WITHIN ABOUT 500 FT OF A VFR KING AIR. THE B767-200 TCAS HAD JUST FAILED A FEW MINS BEFORE.

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CLBING OUT OF JFK TO FL280 AND DIRECT TO RBV. TCAS DISPLAY WENT BLANK AND THEN DISPLAYED A FAULT. I WAS PNF AND BEGAN TROUBLESHOOTING TCAS. CLBING THROUGH ABOUT 16800-17000 FT; ZNY CALLED OUT TFC. I LOOKED UP AND SAW WHAT I BELIEVE WAS A BEECH KING AIR FOR JUST A BRIEF SECOND. HE LOOKED TO ME TO BE ABOUT 500 FT ABOVE AND MAYBE 500-1000 FT LATERALLY FROM US. THERE WAS ABSOLUTELY NO TIME TO REACT. I TOLD ZNY THAT I WOULD PROBABLY FILE A RPT BECAUSE THAT ACFT SEEMED PRETTY CLOSE. ZNY CAME BACK AND SAID IT WAS A VFR ACFT AND THEY WERE NOT WORKING IT. I'M STILL NOT SURE IF THERE IS A SET CRITERIA FOR RPTING A NEAR MISS; BUT THAT'S THE CLOSEST I'VE COME TO ANOTHER AIRPLANE AT THAT ALT. THE KING AIR APPEARED LEVEL AT 17500 FT CRUISING E. ABOUT 10-15 MINS AFTER THIS INCIDENT THE TCAS DISPLAY CAME BACK AND OPERATED NORMALLY FOR REMAINDER OF FLT. IF TCAS WAS OPERATIONAL AT TIME OF INCIDENT; THERE IS NO DOUBT WE WOULD HAVE RECEIVED AN RA. I STILL MADE AN ENTRY IN LOGBOOK. AGAIN; I DON'T KNOW IF THERE IS CERTAIN CRITERIA FOR RPTING A NEAR MISS; BUT I NEVER WANT TO GET THAT CLOSE AGAIN!