From my experience even .323" bullets are often too small for some 8x56 M-Sch barrels. Here are the current CIP dimensions, minimum barrel and maximum jacketed bullet for the three similar cartridges:
8x57 I: bore/land min. .3071", groove min .3177", bullet max .3185"
8x57 IS: bore/land min .3106", groove min .3228", bullet max .3236"
8x56 M-Sch: bore/land min .3130", groove min .3268", bullet max .3248"