We have investigated the effect of midgap bound states on the Josephson properties of symmetric 45°  tilt bicrystal grain-boundary junctions (GBJ’s) characterized by different junction widths. GBJ’s, 1μm wide, show a monotonic temperature dependence of the Josephson current with a slight upwards curvature, the result of the competition between midgap states and the opposite-sign continuous current. In some junctions, an increasing contribution of the midgap-state-mediated current is observed at low temperatures, leading to an anomalous nonmonotonic temperature dependence of the critical current. This behavior is more evident for junctions smaller than 500nm, where, in some cases, a 0- to π-junction transition is clearly observed. Experimental results agree qualitatively with theoretical models taking into account midgap states in strongly nonuniform barrier interfaces.