Walter Cunningham cannot accept the quarter from Miss Caroline because his family would not be able to pay her back. (Hes exremely poor) Scout explains this to Miss Caroline, that he is a "Cunningham" Since Miss Caroline is new to the town she did not understand and Scout explains to her that the Cunningham's don't take anything they can't pay back. (This happens on Page.20)
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Walter Cunningham refuses Miss Caroline's money because his family is too proud to accept charity they can't repay. It goes against their values of self-sufficiency and dignity. Walter's father might not approve of him taking money without being able to return the favor.
Cunningham won't take Miss Carolines quarter because he wouldnt be able to pay it back.