womanwalksalone on Family Tree Circles

After spending years looking for Lark Tate, my great grandfather, I discovered James L. Tate listed on the 1880 census along with his sons (my grandfather's half-brothers) James and William along with his first wife, Martha. I didn't know whether to laugh, cry or yell with joy.

The elation didn't last too long though because in the same district in Pulaski County, Illinois, the same year, I found another Jas L. Tate, same age, listed with his father, William. If anyone out there has any clue as to the truth of "The two James", please let me know.
Did the census taker just happen to find James L. are his parents and count him twice, or are these actually two different people.

Hoping someone can help.