Pete,
There's a very simple optical rule for the potential amount of aiming error with a scope of a certain magnification.
The average human eye (meaning 20/20 eyesight, whether "bare" or corrected) can resolve about 1 inch at 100 yards in a black-and-white test, which involves looking at a sheet with alternating black-and-white lines of equal width. At 100, 20/20 eyesight can see alternatiing lines 1/2" thick, but not very far beyond 100 yards the same lines merge and appear gray (just like a zebra in the distance).
Magnification allows our eyes to resolve B&W lines at longer distances, and also allows us to resolve smaller lines at shorter distances. A 4x scope allows us to resolve 1/4 inch at 100 yards, if the optics in the scope are decent, which means a 4x scope will result in 1/4" of aiming error--in addition to other inherent errors in rifle/ammo accuracy. (This is leaving out parallax, wind, and our hold and trigger pull.)
The same rule applies to higher magnification. A 6x scope allows 1/6" in aiming error (.1666 inch), a 3-9x set on 9x .1111 inch (a little over a 1/10th), and a 25x scope .04 inch.
The quarter-inch error in 4x scopes really isn't much. I have several medium-bore hunting rifles with 4x scopes, and the most accurate will regularly shoot 5-shot groups averaging an inch with the right loads. Which means they should average .85 inch with a 3-9x set on 9x, or slightly over .75" with a 25x. In reality that isn't much error.
I suspect most of the problem many people have with shooting small groups with a 4x scope is due to inappropriate targets. Most commercial targets these days have pretty small aiming points, because so many hunters use at least a 3-9x variable. While a 4x scope ideally results in only 1/4" of aiming error, most aiming points don't allow us to hold as closely as possible with 4x.
There's also potential parallax, which many hunters don't even know how to check for, or correct. But 4x scopes normally don't have any perceivable parallax at 100.
