As a young lad in the 50's, I also saw a few of the .236 Navy rimmed rounds.  My high school English teacher had a second job as the Collectible Cartridge Editor of the old Gun Report, so I asked him.  Here is what I was told:

In the early 1890s, about the same time that the Krag cartridge was being developed, the European military establishments (and Mexico) were investigating very small bore, smokeless cartridges with jacketed bullets.  A lot of experimental work was done in Germany & Switzerland; Bore sizes ranged from 5.5 mm to 6.5 mm.  That is where the 6.5 Mannlicher cases came from (Dutch, Rumanian & Carcano), and also the 6.5 x 55. 

Winchester did most of the developmental work that led to the 6mm Navy cartridge. The original cases were rimmed.  A very few (<5) Winchester single shot actions were used as test rifles for cartridge development.  There was (probably) a Gatling and a Browning "potato digger" machine gun made up for test purposes.  There are also some .236 rimmed cases made by Remington, with different shoulders. Remington was trying to keep on top of the new trends.  Remington made a few of their No. 5 rolling block rifles to test ammunition.

When the USN decided on a rimless case and the straight-pull action, developmental work on the rimmed cases stopped.

Winchester advertised their 1885 High Wall Single Shot and the 1895 in .236 Rimmed, but I think only a few developmental rifles were made.  After the rimless Navy cartridge was standardized, Winchester developed a rimless cartridge extractor for the High Wall and sold a few rifles in 6 mm USN caliber.

The Spanish-American War demonstrated that the 6mm Navy cartridge was out-ranged by both the Krag and the 7x57 Mauser cartridges.  That is why it was abandoned.