A C310 LANDED ON RWY 13 AND DIDN'T TURN OFF AT THE AVAILABLE TXWYS AND WAS ABOUT TO CROSS AN INTERSECTING ACTIVE RWY WHEN HE SAW AN ACR B737 ABORTING HIS TKOF ON RWY 9L. RPTR HEARD CALLS FOR AN ACFT TO STOP; BUT DID NOT THINK IT WAS FOR HIM. TRAINING IN PROGRESS IN THE TWR AND RESPONSIBILITY ACCEPTED BY TWR SUPVR.

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PBI WAS LNDG AND DEPARTING RWYS 9R/L AND 13. LAND AND HOLD SHORT OPS WERE IN EFFECT. WE WERE CLRED FOR A VISUAL APCH TO RWY 13 AND AFTER SWITCHING TO THE TWR WERE CLRED TO LAND ON RWY 13. THE CLRNC WAS READ BACK WORD FOR WORD. THE SUN WAS STARTING TO SET TO THE W WHICH MADE IT DIFFICULT TO SEE ANY POSSIBLE TFC TO THE W (TFC ON RWY 9R/L). WE PROCEEDED TO LAND ON RWY 13; TOUCHED DOWN AND SLOWED THE AIRPLANE TO A FAST PACE TAXI SPD. SINCE WE WERE PAST THE FIRST FEW TXWYS; I DECIDED TO TAXI THE ACFT ACROSS RWY 9L TO THE NEXT TXWY BEFORE WE MADE A R TURN. BY THIS TIME THE ACFT IS AT A SLOW PACE TAXI SPD. IT IS AT THIS POINT THAT TWR CALLS A CERTAIN ACFT WITH A 'HOLD SHORT' INSTRUCTION. THERE WAS NO ACKNOWLEDGMENT. TWR ONCE AGAIN INSTRUCTED THAT ACFT TO HOLD SHORT; STILL NO REPLY. BY THAT TIME I STARTED TO LOOK AROUND AT THE OTHER RWYS WITHOUT SEEING ANYTHING THREATENING. THE SAME CTLR APPARENTLY CLICKED HIS MIKE. THERE WERE OTHER VOICES IN THE BACKGROUND (OTHER CTLRS) THAT WERE YELLING 'HE'S GOT TO STOP;' 'STOP' NEVER GIVING A CALL SIGN. I ONCE AGAIN TURNED TO MY R; AND TO MY SHOCKED SURPRISE SAW A B737 DEPARTING ON RWY 9L. AT THE SAME TIME THE TWR CTLR; POSSIBLY A DIFFERENT CTLR; INSTRUCTED US TO HOLD SHORT IMMEDIATELY. I ACKNOWLEDGED. AT THAT POINT WE HAD ALREADY STOPPED ON THE RWY. THE B737 CROSSED RIGHT IN FRONT OF US; ANNOUNCING THEIR ABORTED TKOF. THERE WAS DEAD SILENCE ON THE RADIO FOR A FEW SECONDS. TWR THEN INSTRUCTED US TO TURN R AT THE NEXT TXWY AND TO CONTACT GND WHICH WE DID. STILL MONITORING GND JUST BEFORE SHUTDOWN; I HEARD THE B737 PLT ASKING FOR AN EXPLANATION AS TO WHAT HAD JUST HAPPENED. THE GND CTLR STATED THAT THE ACFT THAT LANDED ON RWY 13 WAS ISSUED A LAND AND HOLD SHORT CLRNC. THIS WAS NOT THE CASE. THERE WAS NO HOLD SHORT RESTR WHATSOEVER. I MADE SURE THAT THIS WAS UNDERSTOOD BY THE GND CTLR. I WAS GIVEN A TELEPHONE NUMBER FOR THE TWR AFTERWARDS. SINCE I WAS 100% CONFIDENT THAT THERE WAS NO HOLD SHORT RESTR; I DECIDED TO DRIVE OVER TO THE TWR TO TALK TO THE SUPVR. ONCE I WAS LET IN THROUGH THE GATE; WHICH TOOK A GOOD 8 MINS; I WAS MET BY THE SUPVR. I PROCEEDED TO EXPLAIN THE SIT FROM MY POINT OF VIEW AND ALSO MADE CLR THAT I FOUND IT TO BE VERY DISTURBING THAT THE GND CTLR WITHOUT KNOWING ALL THE FACTS MADE THE STATEMENT IN WHICH HE ACCUSED ME OF NOT HAVING HEARD THE HOLD SHORT RESTR. HOWEVER; TO MY SURPRISE; ATC CHANGED THEIR STORY. THE SUPVR INDICATED THAT THERE WAS TRAINING GOING ON IN THE TWR AND THAT THE TRAINEE INFORMED HIM OF THE OCCURRENCE IMMEDIATELY AFTERWARDS. HE ADMITTED THAT THERE WAS NEVER A HOLD SHORT RESTR ISSUED TO ME. I THINK THAT IT IS VERY CLR THAT BOTH PLTS AND CTLRS NEVER STOP LEARNING. HOWEVER; IT IS ALSO CLR THAT THIS LEARNING PROCESS AND ESPECIALLY THE LEARNING ENVIRONMENT MUST ALWAYS BE AS SAFE AS POSSIBLE. SAFETY AND SUPERVISION IN THIS TRAINING ENVIRONMENT WERE NEGLECTED. EVERY PARTY INVOLVED CAN TAKE A DEEP BREATH AND BE GLAD OF THE OUTCOME. IT WILL BE A CONSTANT REMINDER FOR HOW IMPORTANT IT IS TO ALWAYS BE VIGILANT.