One of the earliest and most graphically depicted examples of the so called "Bradley Effect" took place many years before Bradley ran for Governor of California, even many years before he was Mayor of Los Angeles.
It happened in Detroit in the 1940s. It involved a lawyer named Edwards; George, not John.
George Edwards, a labor leader, was running for Mayor against a guy named Cobo, who represented the interests of the local real estate and business owners.
Edwards, who was white (and who would later hold several public offices, including Detroit's City Clerk, and who eventually would serve judge on the U.S. Sixth Circuit Court of Appeals) was backed by organized labor, which (supposedly)controlled a large majority of Detroit's electorate. With labor's support, Edwards was favored heavily to win.
A major component of Edward's campaign platform was to do away with segregated neighborhoods and housing. Cobo(who also was white), vowed to maintain the status quo in Detroit, which consisted of very strictly segregated neighborhoods, both by race and ethnicity.
Because most of the labor unions were integrated, and workers supported integration in the workplace, it was believed that union workers, and leaders, many of whom were precinct delegates and precinct captains would vote for Edwards.
Instead, Cobo (who would die in office, thereby avoiding going to prison for blatant, criminal public corruption) won by a landslide.
Afterwards Labor leaders asked their rank and file members what had happened. Those who would speak said ( paraphrasing) "it was one thing to work side by side with blacks on the assembly line, but something entirely different, and not acceptable to live next door to blacks."
The point is that even back in the 1940s, white voters would say one thing publically when it came to racial issues, but vote entirely different, when they got inside that secret voting booth.
See Segrue's _ Origins of the Urban Crisis_ for more details about the mayoral race between Edwards and Cobo.