Dave Petzal explores this question over at Field and Stream, and comes to basically the same conclusion that I've come to. No, basically, a cartridge is just a container for powder and bullet.
Some cartridges seem to shoot more accurately than others, but that's because some cartridges are made into accurate rifles, and some companies make really fine match components. Careful attention to detail, both in assembling the rifle and assembling the ammo, makes an accurate combination, but sometimes it's impossible to assemble match ammunition from over-the-shelf components. When's the last time you saw match-quality .30-30 brass, or match quality .35 Remington brass? I don't believe that Lapua or Hornady make much of it. On the other hand, match .308 brass is fairly ease to come by, and I'd bet that each and every piece of 6BR brass is match quality. If your brass isn't consistent, you'll probably not get stellar accuracy, although many of us get great accuracy from the .30-30. I've shot some groups with a bone stock M94 that would make tactical competitors proud. I've got a Marlin 336 that will put a 200 grain .35 caliber bullet into two inches all day long at 100 yards, and that with a 2X scope. If I could see the target and had a match barrel on the rifle, it might do very well indeed.
The most important part of accuracy is the nut behind the trigger. If the ammo is carefully assembled and fired through a good barrel, the difference in the target will be the difference between the riflemen. A guy that shoots 1000 rounds a year will probably shoot better than the guy who shoots 20 rounds per year.
In 1989 I found myself commanding a SWAT team as an additional duty at Fort Polk, LA. Our marksman was issued two rifles, one built on a Remington 700 action, and another built on a Winchester 70 action. The shooter's name was Williams. One day we were out shooting and he was making bug-holes at 100 yards with his Remington. He complained to me that the rifle wasn't right. His bug-holes were bigger than they should be. In the nature of SWAT teams, I told him to quit complaining and shoot the equipment he was issued. But, after the exercise, we boxed-up the rifle and sent it to the AMU at Fort Benning. Sure enough, something was wrong with the rifle. They fixed it and sent it back.
The point of that whole story is that Williams was a very good shooter with lots of rounds through his rifle and he knew exactly what to expect from it. I doubt that Williams cared what caliber he was shooting. A good rifle with good ammo and lots of opportunity to practice, and Williams could probably have cared less if he was shooting a .308 or a .30-30.