While I was going back through the information on my mother's side of the family, I solved one mystery: what ever happened to Catherine Rodgers' first husband, James McAneny? She remarried shortly after their last child was born, but I was never able to find James' death record. Why? Because I was looking in Cleveland and he died in Chicago.
Here is an image that was once at Familysearch.org:
James McAneny's death record
So Catherine was a month or so pregnant with James' last child, Lillian, who was born in April 1911. Here is an summary of the information on his death record:
James McAneny, age 38, white male, married, born in Ireland, birth date and parents unknown. Length of residence in US, Illinois, and Chicago all unknown.
Died Aug 12, 1910.
Occupation: Boiler maker, Chicago & Northwest Railway, which he worked at from 1905 to 1910.
Usual residence: 4123 Carroll Ave
Died at Monroe Street Hospital of shock and scalds, suffered while cooking a flue on C & NW Ry Co. Engine 1334. Determined by an inquiry by Peter M Hoffman, coroner or coroner's physician.
Burial: August 15, 1910, in Cleveland
Edited to add: A descendant of Lilian McAneny has confirmed that James died in Chicago. He remembers seeing the telegram that his grandmother Catherine received informing her of the news.