bonefish,
In some instances, an improved cartridge probably can be more accurate than the parent round, due to firmer headspacing on the sharper shoulder. The .22 K-Hornet is an example, because if handloaded correctly it headspaces on the shoulder rather than the rim, and the rims on .22 Hornet brass can vary slightly in thickness. Bullets from a K-Hornet end up a more consistent distance from the lands, which tends to produce finer accuracy.
The same is probably true of some rimless rounds with sloping shoulders. The .300 H&H can be very accurate, but many people report even better accuracy with belted .300 magnums with the brass resized so they headspace on the shoulder. Certainly I've gotten better accuracy from various .300 Winchester Magnums, on average, than the .300 H&H Magnums I've owned and fooled with.
However, to really test whether "improving" results in finer accuracy, it would take a lot of shooting of a lot of rifles. I've owned two .22 K-Hornets, both very accurate, a CZ 527 and my present one, a Brno ZKW 465, the "ancestor" of the 527. Both those rifles are normally very accurate anyway, and I acquired both after they'd been rechambered. Did they shoot more accurately than when they were standard .22 Hornets? Dunno--but I also owned a Ruger No. 1B .22 Hornet that's just as accurate as those two .22 K-Hornets.
For a while I had a NULA in .257 Roberts AI, which was very accurate. But my wife has a NULA in the standard Roberts that shoots just as well.
A sharper shoulder can help some powders burn more consistently, but from what Ron Reiber has observed, a 30-degree does this far better than the 40-degree shoulder of Ackley Improved rounds. Interestingly, the RCBS series of improved cartridges, developed by Fred Huntington, all had 28-degree shoulders. They may have been more accurate than the Ackley Improved rounds, but who knows? They're largely forgotten these days.
Another factor is that a lot of rifles chambered in improved rounds are custom-made, with very good barrels. They're probably more accurate than the average rifle, but how much is due to the cartridge and how much due to the rifle?
