What these white people are saying is that their people generally worked their way up to the level of other whites, so why can't the black community too?
This claim overlooks, of course, the fact that nearly all people who became white "emigrated," a word that means they came by choice. It also trivializes the brutal realities of the Middle Passage and of slavery in America.
Do the whites of today who play this ethnicity card compare the sufferings of their opportunity-seeking ancestors to those of kidnapped black ancestors because the collective white psyche is so repressive and forgetful?
Update: See Richard Jensen on the prevalence of an apparent myth among Irish Americans, that their ancestors commonly faced signs saying "Help Wanted--No Irish Need Apply!":
The Irish American community harbors a deeply held belief that it was the victim of systematic job discrimination in America, and that the discrimination was done publicly in highly humiliating fashion through signs that announced "Help Wanted: No Irish Need Apply." This "NINA" slogan could have been a metaphor for their troubles—akin to tales that America was a "golden mountain" or had "streets paved with gold." But the Irish insist that the signs really existed and prove the existence of widespread discrimination and prejudice.
The fact that Irish vividly "remember" NINA signs is a curious historical puzzle. There are no contemporary or retrospective accounts of a specific sign at a specific location. No particular business enterprise is named as a culprit. No historian, archivist, or museum curator has ever located one; no photograph or drawing exists. No other ethnic group complained about being singled out by comparable signs. Only Irish Catholics have reported seeing the sign in America—no Protestant, no Jew, no non-Irish Catholic has reported seeing one. This is especially strange since signs were primarily directed toward these others: the signs said that employment was available here and invited Yankees, French-Canadians, Italians and any other non-Irish to come inside and apply. The business literature, both published and unpublished, never mentions NINA or any policy remotely like it. The newspapers and magazines are silent. The courts are silent. There is no record of an angry youth tossing a brick through the window that held such a sign. Have we not discovered all of the signs of an urban legend?