Heavier bullets are longer and thus require a faster rate of twist in order to stabilize. Shorter bullets, while stabilized with a slower twist, are still okay from a faster-twist barrel.
There is thus a range of bullet weights which give good groups in any particular twist-rate barrel.
W.E.G.'s comment about thin jackets is appropriate for some bullets in fast twist-rate barrels. There are varmint bullets designed for lower velocities which will indeed come apart with high velocity and fast twist. Those designed for the .22 Hornet, for instance.
1:9 should work just fine for bullets from 40 grains to 70 grains; maybe even a tad heavier, but I'm not experienced there.