DENVER—For Janessa White, the American dream of a red brick house on a tree-lined street blocks from a good elementary school remains obtainable. She just has to rent it.

Ms. White and her boyfriend moved with her 7-year-old son from Missouri to Denver last year. In Missouri, Ms. White owned her home, which she bought for a little over $100,000. To buy a house like the one she rents in Stapleton, an affluent section of the Colorado capital, would cost about four times as much. Even though her household’s income is in the low...