Can't speak for the 6.5x55 as have never handled an early military rifle in that calibre, but the 6.5x55, 7mm and 8mm Mauser, our .303 and the American 30 Kroger, 30-03 and 30-06 all appeared in the last decade of the Victorian era or first few years of the last century. Most had rear sites graduated to 1000 yards and can provide accurate and deadly fire out to that sort of range. Boer marksman caused havoc with their 7mm mausers at very long ranges. As to deadly range that's probably getting on for a few miles.
As as to being able to ensure an accurate one shot kill, well most modern sporting rifles should be able to hold with a deer skill zone out to 400 yards if not a bit more. It is then down to the skill of the stalker to know the trajectory and to read the wind and place that bullet accordingly. That takes an awful lot of skill and time on the range. Most of us run out of skill at 200 or so yards.