For a set $X$, let $2^X$ be the power set of $X$. Let $B_X$ be the Boolean graph, which is defined on the vertex set $2^X\backslash\{X,\emptyset\}$, with $M$ adjacent to $N$ if $M\cap N=\emptyset$. In this paper, several purely graph-theoretic characterizations are provided for blow-ups of a finite or an infinite Boolean graph (respectively, a preatomic graph). Then the characterizations are used to study co-maximal ideal graphs that are blow-ups of Boolean graphs (pre-atomic graphs, respectively).