  We discuss the role of K3 surfaces in the context of Mercat's conjecture in
higher rank Brill-Noether theory. Using liftings of Koszul classes, we show
that Mercat's conjecture in rank 2 fails for any number of sections and for any
gonality stratum along a Noether-Lefschetz divisor inside the locus of curves
lying on K3 surfaces. Then we show that Mercat's conjecture in rank 3 fails
even for curves lying on K3 surfaces with Picard number 1. Finally, we provide
a detailed proof of Mercat's conjecture in rank 2 for general curves of genus
11, and describe explicitly the action of the Fourier-Mukai involution on the
moduli space M_{11}.
