NMAC BTWN A C172 TURNING FINAL IN FRONT OF ANOTHER ACFT ON FINAL.

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WE WERE RETURNING TO CALDWELL AFTER DEPARTING THERE FOR A TRAINING FLT. WE WERE SLIGHTLY S OF LINCOLN PARK ARPT. UPON CONTACTING THE TWR; WE WERE INSTRUCTED TO 'RPT OVER THE LINCOLN PARK WATER TWR; ENTER A R BASE FOR RWY 22.' THE PLT DID NOT UNDERSTAND THE INSTRUCTIONS AS HE WAS NOT FAMILIAR WITH LCL LANDMARKS AND THE LINCOLN PARK WATER TWR IS NOT MARKED ON ANY CHART. SOME CONFUSION AROSE AND WAS DISPELLED BY THE INSTRUCTOR POINTING OUT THE LINCOLN PARK WATER TWR. COMS WERE RE-ESTABLISHED WITH CALDWELL TWR AND WE WERE TOLD THAT WE WERE '#2 ON A R BASE FOR RWY 22.' THE PLT DID NOT IMMEDIATELY SEE ANY TFC BTWN HIS ACFT AND THE RWY. THEN HE SPOTTED AN ACFT ON FINAL AND CALLED THAT HE HAD THE TFC IN SIGHT. HOWEVER; THIS WAS NOT THE '#1' ACFT ON R BASE FOR RWY 22. WHEN WE TURNED FINAL FOR RWY 22; WE CUT IN FRONT OF THE OTHER ACFT. THEY SPOTTED US IMMEDIATELY AND INITIATED A R 360 DEG TURN. WE BOTH SAW THE OTHER ACFT AND HEARD THEM ON THE CALDWELL TWR FREQ. WE THEN INITIATED A 90 DEG R TURN AND DEPARTED THE TFC PATTERN TO REGROUP. AN NMAC HAD OCCURRED. THE HORIZ SEPARATION WAS APPROX 500 FT AND THE VERT SEPARATION WAS LESS; MAYBE 20 FT. THE COMS FROM CALDWELL TWR REGARDING THE LINCOLN PARK WATER TWR CONFUSED THE PLT. THE PROCESS OF DETERMINING WHAT THE TWR HAD SAID AND WHAT IT MEANT CONTRIBUTED TO THE PLT LOSING SITUATIONAL AWARENESS. WHEN HE SPOTTED THE ACFT ON FINAL FOR RWY 22; HE THOUGHT THAT WAS THE OTHER ACFT ON R BASE FOR RWY 22. HE THOUGHT THAT THE ACFT IN FRONT OF HIM HAD TURNED FROM BASE TO FINAL BEFORE HE SPOTTED IT. THE PLT SHOULD HAVE TOLD CALDWELL TWR THAT HE WOULD CIRCLE WHERE HE WAS UNTIL HE WAS COMFORTABLE WITH HIS UNDERSTANDING OF THE OVERALL SIT IN THE CALDWELL TFC PATTERN. THE PLT HAD JUST COMPLETED A TRAINING FLT AND WAS FATIGUED. THE PLT WAS FLYING WITH A FLT INSTRUCTOR AND WANTED TO PERFORM ALL REQUESTED TASKS. THE PLT WAS RELUCTANT TO ADMIT THAT HE WAS FATIGUED AND NOT UNDERSTANDING THE SIT TO THE INSTRUCTOR BECAUSE THAT MIGHT BE VIEWED UNFAVORABLY. THE PLT SHOULD HAVE TOLD THE INSTRUCTOR AND ASKED THE INSTRUCTOR TO TAKE OVER COMS OR TO BREAK OFF THE APCH; SO THAT THE PLT COULD REST AND RE-ESTABLISH HIS SITUATIONAL AWARENESS. WHEN THE PLT ASKED THE TWR CTLR TO REPEAT HIS COMS REGARDING THE WATER TWR; THE CTLR REPLIED IN A MANNER THAT DISCOURAGED FURTHER QUESTIONS. THIS CONTRIBUTED TO THE PLT'S CONFUSION BECAUSE HE WAS RELUCTANT TO CONTINUE TO ASK FOR CLARIFICATION FROM THE CTLR. THE CTLR DID NOT HAVE RADAR CONTACT AND DID NOT HAVE VISUAL CONTACT WITH ALL THE ACFT THAT HE WAS CTLING. IF THE CTLR HAD EITHER RADAR CONTACT OR VISUAL CONTACT; HE WOULD HAVE ISSUED INSTRUCTIONS SO AS TO PREVENT THE NEAR MISS.