Yes, indeed you can. Just keep in mind that the 1600 on the box is not the film's ISO speed. The ISO speed is not stated by Fuji, but most independent sources figure it to be 640-800 in regular b/w developers.
FWIW, none of the super-high-speed b/w films (Fuji Neopan 1600, Kodak T-Max 3200, or Ilford Delta 3200) state the actual ISO film speed on the box. Ilford and Kodak at least state it in their data sheets, but not Fuji.
I shot a roll of 35mm Neopan 1600 (at EI 1600), and processed it in Caffenol-C using the formula on Digital Truth. Time was 15 minutes at ~72 F. The results had LOTS of base fog but I got usable negatives which scanned very well; haven't tried printing them optically yet.
I developed Neopan1600 @ 1600 in Caffenol-C-H, there is some potassium bromide added. Works charmingly with low fog and great tonality. 15 mins development. Details here: http://caffenol.blogspot.com/
Another great developer for Neo1600 @ 1600 is Diafine. Much smaller grain.