WHILE IN LEVEL FLT ON APCH TO CUN ARPT; ACFT X OBSERVES ACFT Y ON TCASII; OPPOSITE DIRECTION. ALTHOUGH THE CTLR ADVISED THE ACFT X CREW TFC WAS NOT A FACTOR; THE PLT BECAME UNCOMFORTABLE WITH THE SIT AND BEGAN AN EVASIVE MANEUVER. WHILE IN A R TURN; THE PLT RECEIVED A TCASII RA.

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WE WERE CLRED TO 2000 FT. APCHING 10000 FT WE LEVELED OFF TO SLOW TO 250 KTS WHEN WE NOTICED AN ACFT AHEAD 20 MI ON TCASII. IT WAS CLBING DIRECTLY TOWARD US. WE QUERIED THE CTLR WHO TOLD US THERE WAS NO FACTOR 20 MI ON TCASII THOUGH IT WAS CLBING DIRECTLY TOWARDS US. WE REPEATED OUR CLRNC. 10 MI AWAY FROM THE INTRUDER; WE TOLD CUN APCH THAT WE WERE MAKING A R; N; TURN TO AVOID TFC. DURING THE TURN WE RECEIVED AN RA TO CLB. THE GREEN BAND ON THE IVSI WAS 5000-6000 FPM. THE CAPT IMMEDIATELY LEVELED OUT AND CLBED THE ACFT REACHING THE GREEN BAND. WE THEN GOT AN 'INCREASE CLB' WARNING. THE THROTTLES WERE ALREADY WELL OVER MAX THRUST. WE OVERBOOSTED THE ENGS AND SEARCHED FOR A VISUAL ON THE INTRUDER. THE FO AND I SAW THE ACFT PASS BY THE R SIDE OF THE ACFT 300 FT BELOW AND APPROX 1.5 MI AWAY. I FIRMLY BELIEVE IF WE HAD NOT TURNED AND CLBED; WE WOULD HAVE HAD A MIDAIR COLLISION. WE INFORMED THE CTLR OF OUR ACTIONS. HE SAID; 'NO PROB; CLRED TO 2000 FT' LIKE NOTHING HAPPENED. SUPPLEMENTAL INFO FROM ACN 400844: THE CTLR IDENTED THE INTRUDER AS A B727 DEPARTING CANCUN AND IT WAS NOT A TFC CONCERN.