Landships II

OK, I'll play "straight man" :).

Believe it or not, Cavalry, Artillery, Engineers, (see also http://landships.activeboard.com/t38854104/p-1903-bandoliers/), Intelligence etc. and, of course, Infantry, were and remain "the arms" or "fighting troops" and would be expected to carry 50 rounds of .303 Mk VII ball (or whatever) in 5 x 2 x 5-round charger clips in those bandoliers. 

Pretty much superseded in front-line use by WW1 but certainly still in use by some fighting corps and formations in some armies (still on issue until 1941 for Aus Inf according AWM but not, I think, front-line).  Corps which descended from those are "the arms" too - Armour, Signals, Army Aviation - except the Electrical and Mechanical Engineers (WW2), they were steadfast in "the services", along with Service Corps, Medical, Ordnance, etc.  Hmmm ... also used in WW1 by the Australian Army Service Corps according to AWM, again.  No "simple" rule then.

I suspect that when such as the Mounted Infantry (for instance the Light Horse Regiment in Australian forces) discarded their leather bandoliers in the trenches for something more practical, the other "arms" snapped them up with glee - and they were probably already part of some official (or superseded) order of dress as shown for the Engineers in that link above. 

More importantly, as you observed, fine for holding (small) packs of cigarettes and posing for photographs.  And, in the latter case, the machine-gunners wouldn't loan them their cartidge belts and Mexico was neutral, so sombreros were "out".

Things change - I suspect the division of "the arms" from "the services" went pretty much by the board with the doctrine of "self protection" in more recent years (1980s-1990s?), FWIW. And perhaps "self protection" is the reality in any war, regardless of doctrine, as shown by the AASC being issued with (but slender evidence thus far - though I don't doubt they posessed the things) P1903 bandoliers.