AN ACR MD80 FLC WAS ROLLING OUT AFTER LNDG ON RWY 4 WHEN THEY REALIZED THAT ANOTHER MD80 WAS ON TKOF ON RWY 31 AND THEY EXPEDITED THROUGH THE INTXN TO AVOID A POTENTIAL CONFLICT. THE OTHER ACFT DID A REJECTED TKOF. AN ACFT FOLLOWING THE LNDG ACFT MADE A GAR.

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LNDG LGA RWY 4 WITH RPTED WX 1800 FT OVCST 8 SM AND BRAKING ADVISORIES. ACTUAL WX WAS 800 FT OVCST 2 SM WITH MODERATE RAIN AND PIREPS OF POOR TO FAIR BRAKING B737. CAPT WAS PF AND CALLED FOR MAX BRAKING. LNDG WAS UNEVENTFUL UNTIL WE APCHED THE RWY INTXN AND TWR CLRED ANOTHER ACR FLT TO TKOF RWY 31. TWR THEN INSTRUCTED THAT FLT TO ABORT THEIR TKOF AND WE COBBED THE THROTTLES TO EXPEDITE RWY INTXN XING TO AVOID POTENTIAL CONFLICT. TELECON WITH TWR SUPVR REVEALED THAT THE TWR CTLR THOUGHT THAT WE ACKNOWLEDGED A HOLD SHORT INSTRUCTION DURING LNDG ROLLOUT. WE DID NOT HEAR THAT INSTRUCTION AND DID NOT ACKNOWLEDGE THAT INSTRUCTION. SUPPLEMENTAL INFO FROM ACN 382930: WINDSHIELD WIPERS ON AND MAX BRAKING SELECTED ON AUTO BRAKES. THE LNDG WAS UNEVENTFUL UNTIL WE APCHED THE RWY 31 INTXN AND TWR HAD APPARENTLY CLRED ANOTHER ACFT FOR TKOF ON RWY 31. I IMMEDIATELY ADDED PWR TO EXPEDITE ACROSS RWY 31 TO AVOID A POTENTIAL CONFLICT. SUPPLEMENTAL INFO FROM ACN 382931: CLRED FOR TKOF; RWY 31 LGA. AT 40- 70 KTS; WE WERE ADVISED TO ABORT. DURING ABORT; ANOTHER ACFT ENTERED OUR RWY. ASKED TWR TO EXPLAIN. TWR HAD CLRED THE OTHER ACFT TO LAND RWY 4 PRIOR TO OUR TAXIING INTO POS AND WE WERE UNAWARE OF LNDG TFC ON THE XING RWY. TWR CLRED US FOR TKOF AND THEN TOLD THE OTHER ACFT; WHICH WAS ON LNDG ROLLOUT; TO TURN OFF AT TXWY P; SHORT OF THE INTXN. THE OTHER ACFT; DUE TO SPD AND RWY POS; WAS UNABLE TO COMPLY. IN ADDITION; DUE TO THE HIGH TASK LOAD NORMALLY ASSOCIATED WITH ROLLOUT; THE CREW DID NOT REPLY AND CONTINUED THROUGH THE INTXN. LAHSO IS NOT AUTH AT LGA AND THE INTRUDING ACFT WAS NOT TOLD TO HOLD SHORT PRIOR TO ROLLOUT. NONE OF THIS WAS KNOWN TO US AT THE TIME OF TKOF. IN ADDITION; THE INTRUDING ACFT WAS NOT VISIBLE UNTIL ACTUALLY ON OUR RWY DUE TO RAIN AND VISIBILITY OF APPROX 1 MI AND DUE TO THE TAILS OF THE SEVERAL ACFT LINED UP ON THE TXWY PARALLEL TO OUR RWY. IN OUR OPINION; HAD WE NOT ABORTED; WE WOULD HAVE HIT THE XING ACFT. WE OBSERVED OUR ALT RELATIVE TO THE INTXN ON OUR SUBSEQUENT TKOF AND; DURING A STANDARD PWR TKOF; DEEMED IT INSUFFICIENT TO CLR THE OTHER ACFT. LATELY; I HAVE HEARD SEVERAL CLRNCS ISSUED DURING ROLLOUT AT MANY ARPTS. I SUGGEST WE ISSUE A MESSAGE ASKING CREWS TO MONITOR AND RPT CLRNCS DURING ROLLOUT; RE-EMPHASIZE THE CRITICAL NATURE OF THE LNDG PHASE OF FLT AND REMIND ATC THAT A CLRNC TO LAND CANNOT BE CHANGED AFTER LNDG. THIS COULD EASILY HAVE BEEN AN ACCIDENT.