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We're not sure about the exact history of this place, but we found rental ads for it in 1906 Chicago newspapers. In 1908 it was the subject of a landmark court case, Hart v. Everhardt; the owner stated the building was worth $85,000 (that's $2,076,000 in 2013 currency). According to the maps on HistoricAerials.com, the Bon Air was still standing in 1962, but had disappeared by the time 1972 rolled around.
We don't know when this picture was taken, but noting the lack of vehicles and the fact that both Leland and Winthrop were dirt roads, we suspect it was quite a long time ago. If anyone has any further information about the Bon Air Apartments, please add it to the comments.