Do you kind of remember how it was, or is the 308 Norma Magnum a new name? Well, it has been around for quite awhile. It was introduced in 1959, and started to show up in the United States within about a year. As the name says, it's a Norma creation, and that means Sweden. At least part of the original intent was to provide a magnum-performance-level cartridge that could be adapted to existing 30-caliber surplus military rifles, such as the Springfield. New rifles of high quality were also made for the cartridge. You hardly see any of them today, even at the largest gun shows.
Norma did a fine job with it. The cartridge case is almost a dead ringer for the 338 Winchester Magnum necked down to 30 caliber; the wildcat 30-338. It's about what the shooting public had expected Winchester to come up with as their "short" 30-caliber magnum, to follow 1958's introduction of the 264 Winchester Magnum and 338 Winchester Magnum. But, along came Norma, which may have doused the original plan for the 30-caliber cartridge in Winchester's short magnum lineup. The short magnums of that period fit into a regular-length long action, like a 30-06, and did not require the extra-length magnum action used for the 300 H&H and 375 H&H. Another cartridge to use essentially the same belted case body is the 7mm Remington Magnum, introduced in 1962 along with the Model 700 Rifle. Incidentally, the first to use that basic case was the 458 Winchester Magnum (1956).
When Winchester came out with the 300 Winchester Magnum in 1963, they took a slightly different route. Still requiring the cartridge to cycle in a full-length action, but wanting more powder capacity than the upstart 308 Norma Magnum had, Winchester hunched up the shoulder to provide more body length, shortened the neck to keep the case from getting too long, and simply seated the bullet deeper; down into the powder, in fact. None of that seemed to hurt anything, but the cartridge didn't really put out more velocity than the 308 Norma Magnum. I've had extensive experience with all three of them (308 Norma Magnum, 30-338, and 300 Winchester Magnum), and the velocities at equal pressure may average about 30 fps higher at the muzzle with the 300 Winchester Magnum with 180-grain and heavier bullets. As far as accuracy goes, all three can be amazing in a high-quality match rifle, and excellent in a hunting rifle. The advantage with the Norma is that remarkably uniform brass. But with proper case selection and preparation (and necking down with the 30-338), there won't be much difference in long-range accuracy. Because of all the fussing with 30-338 cases, my match rifles chambered for it have shot sub-inch at 300 yards when the weather and light conditions were easy. That's five-shot groups.
So, expect the 308 Norma Magnum to do the same job as the 300 Winchester Magnum. Suppose you have a magnum action available; regular length, magnum bolt face. Now suppose you want to step up to the plate and have a fine rifle built around it. Is there any particular advantage in building a 308 Norma Magnum over some other cartridge in a similar league? Probably not. Should you do it anyway? Hell yes. In recent times of in-and-out supplies of 300 Winchester Magnum brass, and several other nearby cartridges such as the 7mm Remington Magnum, and all of the quality variations that seem to go along with running ammunition factories to the limit, Norma continues to quietly manufacture ample quantities of unbelievably consistent 308 Norma Magnum brass. Bless their pea-pickin' hearts. So, get tough. Make a plan. Build a fine long-range rifle, medium-weight barrel or heavy. Let the Norma amaze you. No guarantees, but I'll bet you'll never regret it.