I have found the birth entries for both the boys although neither have been named:
Unnamed Brown (male) born 28/4/1915 King, Washington, parents Harry G. Brown and Mary Higgins
Brown (male) born 26/4/1918 King, Washington, parents Harry G. Brown and Mary Higgins
Sorry Mark - I am with Heywood and MaureeninNY too! Although there are some anomalies regarding Mary's surname and parents surnames on the marriage certificate I am positive it is the right marriage and that 'Jim' is indeed Harry G. because:
Mary Jane Higgins arrival 17/2/1913 shows going to H Brown in Montana
Marriage 4 days later on 21/2/1913 in Montana albeit with surname anomalies for Mary. Clearly this marriage was pre-arranged prior to her arrival.
Birth entries of 2 boys in 1915 and 1918 showing mother as Higgins
Mary dies 1918 (father given as John Higgins, rather than George)
Harry G. remarries to Edith in 1919.
1920 census confirms the boys have the right names
Harry G's WW1 Draft regn. has the same address as you have on the postcard
Harry G. states his father was also a Harry - because of this it's more than possible that he had the nickname 'Jim' which he used as an adult on a day to day basis but for all official purposes his correct name was given (similar scenario with my own father).
Harry G. himself is bit of an enigma but I don't want to get sidetracked with him - suffice to say from official records/census etc. his age varies between 1873 and 1881 although birthday of 1st June remains the same, arrival in US varies between 1903 and 1906. He was naturalized in 1909.
'Jim' is clearly Harry G in my opinion and explains why you have been unable to find a marriage for a Jim/James Brown to Mary Jane Higgins, nor births of 2 sons born to a couple with these names. For whatever reason, Harry G. was known as Jim by the Higgins family documents, Mary Jane was going to H. Brown in Montana when she arrived in US 1913, etc. etc. Too much 'fits' rather than doesn't.
Annette