The method of proving lower bounds by bottleneck counting is illustrated for monotone Boolean circuits. This paper gives another proof of the result of Razborov (1985) and Andreev (1985), that monotone Boolean circuits must have exponential size when solving a problem in NP. More specifically, the paper defines a graph recognition problem called BMS. Any monotone circuit that solves BMS, must contain a quantity of gates that is exponential in the eighth root of the input size. The actual instances of the BMS problem used to prove the lower bound are easy to separate for non-monotone circuits. The proof is self-contained and uses only elementary combinatorics.