PA28 PLT AND FLT INSTRUCTOR HAVE CLOSE ENCOUNTER WITH OVERTAKING TBM ACFT ON FINAL APCH TO A BUSY ARPT.

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THE BACKGND/SIT: I WAS R SEAT ACTING AS CFI FOR A RATED PVT PLT ON A LCL FLT FOR HER CURRENCY/PROFICIENCY. THERE WERE AT LEAST 2 OTHER TRAINING AIRPLANES IN THE PATTERN AT THE TIME WHICH IS TYPICAL AT THIS ARPT (BED). WE WERE R DOWNWIND FOR RWY 29 AND INSTRUCTED TO EXTEND THE DOWNWIND FOR TFC. WE WERE THEN INSTRUCTED TO FOLLOW CESSNA TFC TURNING ONTO A L BASE. WE WERE TO BE #3 BEHIND THE CESSNA. THE PLT PICKED UP THE BASE TFC WHEN WE WERE APPROX 4 MI OUT. I SAW THE TFC WHEN IT WAS ON FINAL. THE PLT MANEUVERED ONTO FINAL BEHIND THE #2 TFC. PLT WAS PROPERLY SLOW (65 KTS) TO MAINTAIN SPACING BEHIND THE CESSNA. BECAUSE I DID NOT SEE THE TFC WE WERE TO FOLLOW ON BASE; I ASKED THE PLT IF SHE WAS SURE OF OUR SEQUENCE. THE PLT THEN QUERIED THE TWR IF WE WERE IN THE CORRECT POS. THE TWR RESPONDED WITH IN THE AFFIRMATIVE. I FELT COMFORTABLE AT THIS POINT THAT THE APCH WAS NORMAL. THE PROB: AN AIRPLANE THEN PASSED US ON FINAL ABOUT 500 FT TO OUR R AND ABOUT 150 FT ABOVE US. ITS SPD WAS APPROX 30-40 KTS FASTER THAN US. I THEN HEARD ON THE FREQ FOR THE FIRST TIME; I THINK; THE TBM QUERYING TWR ABOUT THE TFC IT WAS SUPPOSED TO FOLLOW. CORRECTIVE ACTION: MY PLT QUICKLY ASKED THE LCL CTLR WHAT TFC WE WERE TO BE FOLLOWING. THE CTLR SEEMED SOMEWHAT RESIGNED TO THE SITUATION AND TO THE BEST OF MY RECOLLECTION SAID SOMETHING ABOUT IT BEING RATHER JUMBLED OUT THERE ON FINAL. THE TBM WAS INSTRUCTED TO FOLLOW THE CESSNA AND WE WERE TOLD TO FOLLOW THE TBM. MY PERFORMANCE: I RELIGIOUSLY LOOK OUT ON THE FINAL APCH PATH AND CHK THAT IT IS CLR BEFORE THE BASE TO FINAL TURN EVEN AT TOWERED ARPTS. THIS TIME I DIDN'T DO THAT. WE WERE SO FAR OUT ON EXTENDED DOWNWIND AND FOCUSED ON IDENTING THE TFC IN FRONT OF US THAT WE WERE TO FOLLOW THAT IT DID NOT OCCUR TO ME. I MAY NOT HAVE SEEN IT ANYWAY; BUT IT BUGS ME THAT I FORGOT TO FOLLOW MY OWN PROC. I WILL NOT FORGET TO CLR THE FINAL EVEN AT CTLED ARPTS AGAIN. THIS HAD MIDAIR POTENTIAL THROUGH NO FAULT OF OUR OWN.