Breadwinners by class
According to the third row in this table from the American Enterprise Institute (they credit the Census Bureau but I could not find the same indicators in their data)



your hypothesis is incorrect: the Mean Number of Earners per Household is increasing in income for each Quintile. 

I am surprised by the average number of earners in the highest quintile, but I think it is not very surprising that the poorest people are poor because they have trouble finding stable employment.