In W.E.B. Du Bois's seminal work The Souls of Black Folk, he introduces the related topics of double-consciousness and the metaphor of the veil. According to Du Bois, black people in the U.S. are aware of two distinct identities within themselves. One is the "American" identity, which in America is assumed and implied to be white. The other is the "black" or "African" identity. For Du Bois it was impossible to imagine either fully assuming the "White" identity OR to make American identity fully black. It is the struggle to reconcile these two irreconcilable identities that defines the struggles of Black people in America.

The veil refers to several things. Most directly it is the black skin that marks black as different from whites, but it is also the inability of both black and white people to see black people as like white people, as truly American.

One way that some black people have tried to overcome the problem of the veil is by closely adopting the dominant, white culture. Others have rejected that approach. But yes, the difference between the white "American" identity and "Blackness" has long been understood.

I hope that helps answer your question. There is also, of course, the issue of "passing" as white, which some more phenotypically "white" "black" people have tried to do at various times. It has been represented fictionally in Faulkner's Light in August and Roth's The American Stain, just off the top of my head.

I'm going to answer your question in a sideways fashion. During the period you mention, a common expression for an upstanding member of the black community was a "race man".

A race man was someone who held a steady job, dressed well, went to church every Sunday, volunteered for community organizations, and provided for his family.

But this wasn't considered "acting white" - in fact, it was viewed as a matter of racial uplift. In the face of derogatory stereotypes about African-Americans - that they were lazy, prone to criminality, sexually promiscuous - conducting yourself in a morally upright way was an act of defiance, a statement that blacks were just as good as white people.

What's more, race men always spoke up for the interests of their community - they resolutely opposed the injustices African-Americans had to endure under segregation, and did everything in their power to promote black institutions.

The epitome of the "race man" was Booker T. Washington: someone who chose not to directly defy Jim Crow, but instead sought to "uplift" black people through education and hard work.

In retrospect, these men may seem to have been "acting white", but in the context of their time, they were black nationalists: i.e., they promoted self-reliance and institutions run and supported by African-Americans.

I'm not sure if your grandfather fit this description, but it was a common ideal under segregation - sort of the equivalent of the Jewish notion of the "mensch".

For more on the term: https://www.nytimes.com/books/first/c/carby-race.html

http://www.washingtonpost.com/wp-dyn/content/article/2007/12/21/AR2007122101393.html

And on Booker T. Washington: http://www.newrepublic.com/article/books-and-arts/the-race-man