This paper explores analytically the connections between three commonly used statistical measures of agreement: sensitivity, specificity, and Cohen's (unweighed) kappa, which are employed often in clinical studies. In particular, we show that if both sensitivity and specificity are below 7/8, then a kappa of 3/4 or greater (which is considered to indicate good to excellent agreement) can never be achieved even for raw agreement approaching 100%. In addition we derive some of the known properties of kappa for two-by-two tables which apparently are not readily available in the literature.