Well, Winchester did come out with the .405 Winchester which saw reasonable use in both Africa and Asia (in the Model 1895). It was used by the following hunters:
Africa
-President Theodore Roosevelt (although his main elephant and Cape buffalo rifle was his .500/450 Nitro Express Holland & Holland Royale sidelock ejector)
-Charles Cottar (who was actually killed by the cow rhino which he failed to stop with this caliber)
-Jan Lediboor (who was actually killed by the bull elephant which he failed to stop with this caliber)
-Osa Johnson & her husband Martin Johnson (who possessed heavier rifles in their battery)
Asia
- The Marquis De Monestrol
- Kenneth Anderson (although his main rifle for hunting rogue elephants was a .450/400 Nitro Express W.J Jeffery boxlock ejector
A poor sectional density (300Gr bullets of .411 caliber) coupled with anemic velocities (2204fps) and the relatively flimsily constructed nickel jacketed bullets of the era led to the original decline of popularity of this caliber. Then, the original rifles also had an annoying tendency to jam on you, if you loaded the magazine to full capacity and then worked the lever very fast between shots. This eventually caused the original Winchester company to cease manufacture of rifles in this caliber by 1931 and ammunition of this caliber by 1936. Kynoch kept loading 300Gr soft point cartridges until 1957, but they ceased manufacture as well. Today, Miroku manufactures an updated version of the Model 1895 for Winchester. And modern handloading techniques coupled with strongly constructed monometal bullets (as formerly manufactured by North Fork) have brought about a reasonable improvement over the original ballistics of the .405 Winchester.
There have also been a couple of large bore wildcat calibers designed for lever action rifles over the years (with mixed success). When Col. Charles Askins hunted in Vietnam in 1956 (when he had gone there for 12 months to act as a consultant for the South Vietnamese army), he had brought along a Winchester Model 71 that was rebarrelled to a wildcat caliber which he called the .450 Alaskan (400Gr Barnes Original copper jacketed soft points and solids backed by 63Gr of #4895 propellant). It ended up performing very poorly in the field and Col. Askins quickly abandoned it in favor of his .458 WInchester Magnum Model 70 Super Grade.
There is also the .45-90 Express wildcat caliber and the .50 Alaskan wildcat caliber. The .45-90 Express is a modern day modification of the .45-90 Sharps cartridge but loaded with smokeless powder. It can throw 400Gr bullets at 2130 fps. The .50 Alaskan can throw 450Gr bullets at 1718 fps. For the weight and caliber of bullet, these velocities are a little on the lower side (in my personal experiece) for hunting thick skinned African game.
As a final observation, it should be noted that all of the above mentioned calibers have relatively unsatisfactory ballistics for African dangerous game (compared to calibers traditionally used in bolt action rifles). By this, I refer to sectional densities and case capacities.
.405 Winchester: A .411 caliber rifle should be able to throw a 400Gr bullet at 2150 fps (minimum)
.450 Alaskan & .45-90 Express: A .458 caliber rifle should be able to throw a 450Gr bullet at 2150 fps (minimum)
.50 Alaskan: A .510 caliber rifle should be able to throw a 535Gr bullet at 2150 fps (minimum)
This leads me to believe that lever action rifles are ill-suited for high pressure cartridges. I'll also add that the vast majority of lever action rifle designs feature tubular magazines which make the use of pointed nose bullets in them (like the Barnes TSX) unsafe. And then, there is the factor of lever action rifles being not as accurate for long distance shots as bolt action rifles.
