We derive a simple proof, based on information theoretic inequalities, of an upper bound on the largest rates of $q$-ary $\overline{2}$-separable codes that improves recent results of Wang for any $q\geq 13$. For the case $q=2$, we recover a result of Lindström, but with a much simpler derivation. The method easily extends to give bounds on $B_2$ codes which, although not improving on Wang's results, use much simpler tools and might be useful for future applications.