FAILED TO MAINTAIN ASSIGNED ALT TO ANVIL RESULTING IN LESS THAN STANDARD SEPARATION WITH ACFT ON MISSED APCH BFI.

Narrative

SEA WX RPTED AS CLEAR; 15 MI VIS; FOG RESTRICTING VIS TO N AND S. BOEING FIELD; JUST N OF SEA WAS 0-0 IN FOG. HOWEVER; PVT ACFT WERE STILL ATTEMPTING ILS APCHS WITH NO LUCK LNDG AND ALL WERE MAKING MISSED APCHS. F/O WAS FLYING THE ACFT ON AUTOPLT. FLT CONDITIONS WERE VMC ABOVE LARGE AREAS FOR GND FOG. APCH CTL CLRED US FOR AN ILS RWY 16R APCH; MAINTAIN 3200' TO ANVIL; CALL THE TWR AT PARKK. HOWEVER THE PART; 'MAINTAIN 3200' TO ANVIL' DIDN'T SINK IN. F/O THOUGHT IT WAS TO MAINTAIN 3200' TILL ESTABLISHED ON THE ILS LOC. I JUST DIDN'T GIVE IT MUCH THOUGHT. HE'S A SHARP PLT. F/O DSNDED TO 2000' PER THE APCH PLATE. APCH CALLED TO CHK OUR ALT. HE SHOWED US AT 2000'. I ADVISED THAT WE WERE AT 2000' AND IN VISUAL FLT CONDITIONS. APCH TOLD US TO CLB IMMEDIATELY TO 3200' AS HE HAD A MISSED APCH AT BOEING FIELD. HE THEN DIRECTED US TO CLB QUICKLY AND THEN TURNED US 100 DEGS TO THE LEFT AND CLB TO 5000' FOR VECTORS TO ANOTHER 16R ILS. NEXT APCH WAS NORMAL OTHER THAN HAVING TO SIDESTEP TO RWY 16L IN MARGINAL VIS JUST N OF THE ARPT. HOW TO PREVENT THIS? 1) FLY THE CLRNC EXACTLY; 2) CHANGE THE FAR'S TO REQUIRE ALL ACFT TO OPERATE UNDER THE SAME RULES RE: INITIATING IFR APCHS INTO BELOW MINIMUMS WX CONDITIONS.