Classical Brill–Noether theory gives an elegant classification of linear systems on generic curves; the problem of doing the same for higher rank vector bundles has been extensively studied but the picture is far from complete. In joint work with G. Farkas, we prove a conjecture of Mercat for rank 2 bundles on generic curves classifying minimal slope bundles that admit a prescribed number of sections. The proof involves specializing to curves on K3 surfaces and studying the Lazarsfeld–Mukai bundle associated to higher rank bundles. As in the classical case, the Brill–Noether theoretic behavior of rank 2 bundles can be packaged into a “rank two Clifford index; the resulting stratification of $M_g$ is different from the gonality stratification, but we nonetheless show that the divisorial strata in both cases have slope $6+12/(g+1)$.