PLTDEV LTSS BTWN ACR X AND ACFT Y WHEN ACR X CLBED IN PROX TO ACFT Y AT FL280.

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ENRTE LGA TO ORD FL250; CLRED TO FL290 IN 3 MINS OR LESS. THE CAPT COMMENCED A 1500 FPM CLB AND SLOWED THE CLB TO 1000 FPM AFTER 1 MIN. CLB RATE WAS INCREASED TO 1500 FPM AND A TURN OF 20 DEGS TO THE L WAS GIVEN BY THE CTLR. THE TCASII CALLED OUT TFC AND SHORTLY THEREAFTER COMMANDED DSCNT WHICH THE CAPT INITIATED. ATC WAS NOTIFIED OF THE TCASII ALERT AND DSCNT. ONCE CLR OF TFC THE CAPT INITIATED A CLB AND WAS GIVEN A TURN BACK TO COURSE. DATE SEPT/THU/95; ALT FL270; POS 40 NM E OF GIJ. SUPPLEMENTAL INFO FROM ACN 316361: I WAS GIVING THE CAPT A LINE CHK AND I WAS SEATED ON THE JUMP SEAT. WE WERE LEVEL AT FL250 AWAITING FURTHER CLB WHEN ZAU CALLED TO ASK IF WE COULD BE THRU FL290 IN 3 MINS OR LESS. THIS OCCURRED AT XX25 CDT; APPROX 30 MI E OF GIPER VOR. THE CREW RESPONDED THAT THEY COULD MEET FL290 IN 3 MINS AND WERE GIVEN THE CLB CLRNC. THE CAPT BEGAN CLB USING VERT SPD OF 1500. DECREASED TO 1000 AND THEN WAS ABOUT TO INCREASE VERT SPD WHEN ATC GAVE 20 DEG L TURN AND TCASII GAVE TA. THE TA BECAME A RA DSND WHICH WAS FOLLOWED. ATC ASKED IF EXPEDITING TO FL290 AND FO ADVISED IN RA DSCNT. TCASII GAVE CLR OF TFC AND CAPT BEGAN CLB AND FO ADVISED NOW RETURNING TO CLB AND ATC GAVE VECTOR BACK TO COURSE. ATC DID NOT ADVISE THAT CONFLICT HAD OCCURRED. DEBRIEF OF CREW INCIDENT AS FOLLOWS. CAPT DID NOT START CLOCK TO TIME; FO WAS SLOW TO START CLOCK BUT DID. CAPT COULD HAVE USED HIGHER VERT SPD INITIALLY TO EASILY COMPLY. BANK ANGLE IN TURN REQUESTED BY ATC COULD HAVE BEEN INCREASED. FO SHOULD HAVE ADVISED ATC SOONER OF ACTIONS TO FOLLOW RA DSND. SUPPLEMENTAL INFO FROM ACN 316491: ACR X WAS ISSUED EXPEDITE THROUGH FL290 TO MAINTAIN FL330. ACR X SHALLOWED HIS CLB AT FL269 APPROX 1 MIN BEFORE SEPARATION WAS LOST.