This thread is focused on the 1903 pattern bandoleer as used for .30 rifle cartridges from 1903-1918. Due to the relation of the 1909 bandoleer it is included, but the focus is not on the 1909 or latter bandoleers. They are included strictly for discussion purposes.
Prior to 1903 the US Army did not issue small arms ammunition in bandoleers. Ammunition came from Frankford Arsenal (and contracted commercial firms) in 20 round pasteboard boxes, which were then packed in wooden crates lined with zinc to seal against humidity and water damage. In practice a soldier would open a 20 round box and then load his looped Mills cartridge belt, opening more boxes until his belt was full (if ordered to carry a full load). In 1903 the new Springfield magazine rifle (M1903) was adopted and this coincided with new developments in modernizing the US Army. One such development was an experimental cloth bandoleer to carry the new rifle's clipped ammunition. In addition to making resupply and reserve ammunition easier to carry this was likely spurred at least in part by the difficulty in packing clipped ammunition in the pasteboard boxes than an attempt to match the equipment of other nations.
Bandoleers would be opened under order of the commanding officer, the ammunition would then be used to fill a cartridge belt. If additional reserve/emergency supplies of ammunition were needed two bandoleers would be issued per soldier. The ammunition in the bandoleers was to be expended before that carried in the cartridge belt.
Two 1909 pattern bandoleers being worn in conjunction with a cartridge belt and pack: