They almost did it with the .33 BSA. They just never loaded it with heavier bullets for some odd reason. It even has the belt and the bullet diameter is .338 inches!

.33 BSA
"The .33 BSA or .33 Belted Rimless Nitro Express, was developed by necking-down and shortening the .40 BSA. It was introduced in 1921 for the P-14 Enfield Sporting Rifle, catalogued as the BSA "High Power" rifle at the time. The projectile was only 165 gr over 60 gr cordite, and was a bit light for caliber which limited popularity in the hunting field. Factory loads were still listed by ICI up until 1955.
According to the dimensions in Fleming's book, the rim is very slightly rebated."
http://forums.nitroexpress.com/showflat.php?Cat=0&Number=368164

".33 BSA, 26" barrel, no S/N. Blued finish, checkered walnut pistol-gripped stock with checkered forend. Two barrel mounted Express sight-leaves for 150 & 300 yards and two receiver-mounted aperture leaves for 150 & 300 yards, ramp front sight. Rubber butt pad, sling swivels. A sporterized version of the Pattern-14 British service rifle as modified by BSA."

Source of the previous three images: https://www.bidsquare.com/online-auctions/cowans/bsa-33-1923-sporting-rifle-918205

Cheers! Bob F.