.308's and other cartridges in this class (7x57 & 8x57 Mauser, .303 British, .30-'06, etc.) have taken innumerable elephant, rhino, buffalo, etc. in Africa. (As a matter of fact, there are two documented cases of elephant being killed with a .22 LR!!! No, I'm not joking!) Poachers in Africa often use the AK-47's 7.62x39 cartridge, killing anything and everything with it, but they typically fire full-auto.
The big thing is, these cartridges simply don't have enough "punch" to turn or stop a charge from something determined to convert you into fertilizer for the African landscape. Given a clean shot at a vital part, with the animal unaware of your presence, they'll take anything, anytime, with no problems. However, if there is a chance that the animal will become aware of you (before or after the shot) and want to discuss the matter at rather closer quarters, they won't prevent it from getting to you and turning you into something that would gag a gargoyle (to quote C.R. Sam!). They also lack sufficient energy and momentum to penetrate full-length through a major critter, if that's the only shot you have available.
The heavier rifles were designed to deliver enough "punch" to put the animal down in a dangerous situation - and even they don't always work... There's a picture on the wall of a training station in Skukuza, the headquarters of the Kruger National Park in South Africa. It shows a twenty-foot circle of red mud. On one side of the mud is a pair of boots, with a shin-bone sticking up out of the left boot. On the other side of the mud is a hat, upside-down, with neck vertebrae sticking up from it. The mud in the middle is the rest of the hunter... He whacked a buffalo with three rounds of heavy-caliber rifle fire, and fatally wounded it: but the buffalo had enough energy and hatred left in it to reach him and render him into mush for the ants. In another well-documented case, a buffalo absorbed ten solid hits from a .460 Weatherby - over 70,000 foot-pounds of energy, accurately placed - on the run, before finally stopping and expiring. Impressive, no?
Never trust ANY rifle caliber to do the job 100% of the time - it's the shooter, bullet placement and performance, and the mood of the animal that count!