FLT INSTRUCTOR AND STUDENT RETURNING TO DPA ENTERING A L BASE HAVE A CLOSE CALL WITH AN SMA ACFT ON R BASE.

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AFTER DEPARTING DPA TO THE N; WE DECIDED TO COME BACK TO THE ARPT DUE TO DETERIORATING WX. WHEN WE WERE ABOUT 10 MI N WE CALLED DPA TWR WITH INTENTIONS TO COME BACK TO LAND. TWR CLRED US TO RPT 3 MI N FOR A L BASE FOR RWY 15. WHILE INBOUND WE HEARD ANOTHER ACFT CLRED FOR A R BASE FOR RWY 15. ABOUT 3 MI N OF THE ARPT THE CONFUSION BEGAN. TWR TRIED TO WARN US OF THE OTHER TFC; WE LOOKED; BUT NEVER VERIFIED. SO I AM GUESSING WE WERE ALMOST ON PARALLEL FLT PATHS. BUT THEN DPA TWR TOLD US TO TURN W TO AVOID TFC; BUT THE OTHER TFC WAS TO THE W. TWR TOLD US TWICE TO TURN W; BUT WE THEN QUERIED TO TURN E TO AVOID THE TFC AND THEN TWR TOLD US THAT IS WHAT THEY SAID TO TURN TO. SO WE TURNED E AND CAME BACK TO LAND. HOW CLOSE OUR AIRPLANES CAME IS UNKNOWN; BUT WE DID TAKE EVASIVE ACTION BECAUSE WE DON'T SEE THE OTHER ACFT. CONTRIBUTING FACTORS: DETERIORATING WX; PLT TRAINING IN BUSY AIRSPACE; ATC VECTORING; MISCOM. ALTHOUGH I REALIZE HOW HARD IT IS FOR ATC TWRS TO ESTIMATE DISTANCE OF ACFT BUT I THINK FOR THIS SIT ATC COULD HAVE CLRED US OR THE OTHER TFC FOR A 360 DEG TURN FOR SPACING. BUT DETERIORATING WX (FYI; THE FIELD WENT IFR JUST AFTER WE LANDED; SNOW HAD JUST MOVED IN). ALSO MAY HAVE CAUSED THE CTLR TO TRY AND RUSH US BOTH IN. ALL IN ALL; I DON'T BELIEVE WE WERE CLOSE TO A MIDAIR AT ALL BUT THIS MAY HELP CTLRS AND THE CLRNCS THEY ISSUE.