To find the number of cases Jerry hears per hour, we divide the number of cases he hears in $\dfrac{2\dfrac{3}8\text{ hours}}{\text{hour}}=\dfrac{19}{8}$ hours by the number of hours it takes to hear those $19/8$ hours$.\begin{array}{c|c c} & & \\ [-2.5ex] & \hspace{15mm} & \\ [-2.25ex] \textbf{Everett's hours of cases} & \textbf{hours needed} & \textbf{Cases he hears} \\ [0.5ex] \hline & & \\ [-1ex] \dfrac{19}{8} & 1 & 5 \\ [1ex] \dfrac{11}{8} & \dfrac{1}{2} & \phantom{0}2 \\ [1ex] \dfrac{3}{8} & \dfrac{1}{6} & \phantom{0}1 \\ [1ex] \end{array}$In $\dfrac{19}{8}$ hours, Jerry hears $\boxed{8}$ cases.
