I think they are right - it is probably "Imitation of Life" - I found this on the internet for you:
http://us.imdb.com/title/tt0052918/plotsummary
Plot Summary for
Imitation of Life (1959)
In 1947 at Coney Island, down-on-her-luck actress Lora Meredith and her young daughter Susie meet coloured Annie Johnson and her daughter Sarah Jane. Annie is desperate for a place to live and offers to work as Lora's maid for food and lodging. Lora's luck begins to change as a result of their meeting and their two stories unfold across the years. However, Annie's problems start when Sarah Jane tries to pass as white.