A similar question was asked a while back. There were, in the Revolutionary War, some claims for incredible accuracy of American riflemen, and more claims of incredible accuracy many years after the War was over. But it may be assumed that these are questionable ( what? could it be that guys would actually lie about how well they hit a target? Really???). So, an invaluable one is the account of the British officer , Maj. George Hanger , who ( as I posted earlier here )

recorded a shot being made at he and (the much hated) Lt Col. Banastre Tarleton by an American rifleman in 1777. He'd shot in the prone position, and had aimed between them, likely hoping for a drift of wind to carry the ball to one or the other, but there was no wind and instead it passed between, killing the horse of the man behind them. Hanger actually measured the distance afterwards and found it to be 400 yards. When he was later captured, he heard from American riflemen that they could all hit a soldier's head at 200 yards, and given the previous near miss, he believed them.

At 400 yards, the ball ( say, 54 caliber) would have dropped several feet. The typical sights found on a typical American rifle would be low, non-adjusting open sights, and so compensating for the drop would have been much harder than with modern elevating aperture sights. But if that rifleman had spent a lot of time shooting that one rifle, he could have gotten pretty good at estimating what it could do, and obviously he was.

But another precaution is in order, here: when Hanger's riflemen said they could hit the head of a man at 200 yards, consider the apparent size of someone's head at 200 yards. Given average eyesight, those riflemen couldn't likely see any finer detail than a man's head at that distance. Even a very accurate rifle will only shoot as well as the shooter can see.