CL60 AND FA20 EXPERIENCE CLOSE ENCOUNTER AT FL260.

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CRUISING AT FL260; TFC WAS CALLED OUT TO US AT 12-1 O'CLOCK POS AND 10 MI. WE CALLED 'LOOKING' AND LOOKED BUT DIDN'T SEE TFC. WE LOOKED DOWN TO ADJUST OUR MFD RANGE TO LOCATE TFC ON TCAS (IT SHOWED THE OTHER ACFT AT 1 O'CLOCK POS AND +10; DON'T REMEMBER EXACT RANGE) WHILE ATC CALLED OUR POS TO THE OTHER ACFT. AFTER A FEW SECONDS THE OTHER ACFT RPTED HE HAD US IN SIGHT. A FEW SECONDS AFTER THAT; WE BOTH LOOKED UP TO SEE AN ACFT (FALCON 20; WHITE WITH RED OR MAROON MARKINGS) AT OUR 2 O'CLOCK POS; CO-ALT; NOSE-ON; VERY CLOSE (LESS THAN 1 MI). AT THAT SAME INSTANT THE OTHER ACFT MADE A VERY AGGRESSIVE MANEUVER IN BOTH ROLL AND PITCH AND DISAPPEARED BEHIND AND ABOVE OUR ACFT; MISSING US BY MAYBE 500 FT. I THEN INFORMED ATC WE HAD AN NMAC AND AFTER A FEW RADIO CALLS WITH US AND THE OTHER ACFT IT WAS MADE APPARENT THAT SOMETHING HAD INDEED HAPPENED; ALTHOUGH THEIR EQUIP AND OUR EQUIP SHOWED EVERYONE WAS WHERE THEY SHOULD BE. ATC GAVE ME A NUMBER TO CALL AND THEN HANDED US OFF TO THE NEXT CTLR. THE REST OF THE FLT WAS UNEVENTFUL. I CALLED ATC LATER AND THEY WERE STUMPED AS TO HOW THIS COULD HAVE HAPPENED; AND INFORMED ME THAT THE OTHER PLT TOLD THEM THAT THEY WOULD HAVE THEIR EQUIP CHKED. THE TIME BTWN THE INITIAL TFC CALL AND THE NMAC FELT LIKE 20-30 SECONDS. WE RECEIVED NO TCAS TFC ALERT OR RA COMMAND. OUR ALTIMETERS WERE ALL SET TO 29.92 AND WERE WITHIN 50 FT OF EACH OTHER. I ALSO CHKED THEM ON THE GND AND THEY ALL LOOKED GOOD. I SUSPECT THE OTHER ACFT'S EQUIP WAS FAULTY. WE SIMPLY HAD NO TIME TO REACT; AND WE BOTH CAME AWAY WITH THE REALIZATION THAT WE SHOULD NEVER TOTALLY TRUST THE EQUIP AND VOWED TO BE MORE VIGILANT IN OUR TFC SEARCHES FROM NOW ON. WE ARE BOTH EXTREMELY THANKFUL THIS DID NOT TAKE PLACE UNDER IMC.