While the date of death is within the time frame James' parents are said to have died, (while he was in school), Lexicon brings up an important fact, Jo herself said his parents were old when they had him. 57 is not old by wizarding standards.
Dorea (Black) Potter (1920-1977)
Daughter of Cygnus and Violetta (Bulstrode) Black (BFT).
Three older siblings: two brothers Pollux (1912-1990) and Marius (disowned, dates unknown), one sister Cassiopeia (1915-1992) (BFT).
Married to Charlus Potter; they had one unnamed son (BFT).
It is doubtful that Dorea is Harry's grandmother because Jo has told us that "James's parents were elderly, were getting on a little when he was born, which explains the only child, very pampered, had-him-late-in-life-so-he's-an-extra-treasure, as often happens, I think. They were old in wizarding terms, and they died." (TLC) Dorea was only 57 when she died.