Ask yourselves were in the pecking order the fish with the black spots are. Most of the time they are at the bottom.
If there is also a preditor in the tank such as a Dim. Compressicep or Nim. Fusco then the smaller fish in the tank become a possible meal, at least thats what they think.
A small tank will also do it to.
In my own experience, when i first started out with cichlids i stocked all sorts of things in the one tank such as peacocks, red empress, small electric blues x2 and you guessed it a male Dim. compressicep which was an awesome fish, one that i loved to death but as he matured he became extremely aggressive, stalking and hunting everything. The smaller fish were running scared.
It got to the point were three of my fish including 2 peacocks and the electric blue were covered in black spots. So i got rid of Dim. Compressicep and without a doubt within 2 months all fish were back to there normal selves without a single black spot them.
In saying that i have heard that spiralina can cause this occurance also. Try a few different things like diet or maybe remove the into a tank of it's own, see if that get rid of it.
Some fish just don't suit the environment