NASA T-38, N920NS, WAS INBND (HDG EAST) ON A FINKS ONE ARRIVAL TO ANDREWS AFB WITH CLNC TO DSCND TO 8000'. IN THE SAME AREA, PAN AM FLT 140 (AIRBUS A310/N806PA) WAS HDG NORTH AT 7000' AFTER DEPG DULLES (IAD) IAP. THE AIRBUS WAS HANDLED BY IAD APCH CTL. THE T-38 WAS HANDLED BY WASHINGTON (DCA) APCH CTL. THE T-38 PLT DSCNDD THRU 8000' & RPRTD AT 7000'. AT 1ST THE DCA CTLR DIDN'T REALIZE THE T-38 WAS AT THE WRONG ALT. SHORTLY THEREAFTER, THE CONFLICT ALERT ALARMED & THE DCA CTLR ISSUED A TFC ADZY. THE T-38 PLT IMMEDIATELY CLIMBED TO 8000', BUT BEFORE HE CLIMBED, RADAR DATA INDCD THE AIRBUS & T-38 PASSED EACH OTR WITH 711' LATERAL CLNC & 100' VERTICAL CLNC. THE AIRBUS PLT STATED HE SAW THE T-38 PASS AHEAD FM LEFT TO RGT. THE IAD CONTROLLER DID NOTICE TFC FOR THE AIRBUS, BUT HE DID NOT ISSUE A TFC ADZRY. ALSO, HE SAID HIS CONFLICT ALERT SYS DID NOT ALARM; COMPUTER DATA INDCD THE ALARM HAD OPERATED.