Your problem is that the check you add in the end, (s (:, j)) ~ = 1, examines if the sum in column j is more than 1 - but the index in which you add your 1 is a random index between 1 and 15. So the event you wish to avoid can very easily happen in column 14 while you are checking column 1. In addition, because your while loop runs until a mistake is found and then stops - it does not correct the mistake that caused it to exit.
Here is my suggestion:
w = [3,2,4,6]; %% quantities of elements
zw = sum(w);
s = zeros (numel(w),zw); %% initial "empty" matrix
for k=1:numel(w)
for j=1:zw
while sum(s(k,:))<w(k) && sum(sum(s))<zw %move the third condition down
rando=randi(15); %log which index you insert the most recent 1 in
s(k,rando) = 1;
if sum(s(:,rando))~=1 %if that most recent 1 caused your column sum to exceed 1, undo it.
s(k,rando)=0;
end
end %and the while loop will keep running, until you have w(k) 1's in the k'th row.
end
end
