As @GershonGold mentioned in his answer, Mincha time of Shabbat is associated with the passing of Moshe, Yosef, and David. One of the ways we commemorate it is by not greeting someone by wishing them a Good Shabbat at Mincha time. [The Mishna Berurah S"K 6 on Shulchan Aruch (Orach Chaim 292:2) tells us that this is why we say Tzidkatcha after the Amidah in Mincha on Shabbat]
The Rema, in Shulchan Aruch Orach Chaim 292:2, says that for this reason the Ashkenazic custom is not to learn between Mincha and Maariv on Shabbat afternoon. The Mishna Berura (S"K 8) says that this only applies when Mincha is prayed close to the end of the day, otherwise it is not a problem. (He then goes on to describe the limitation of this practice, and the fact that we don't follow it these days).
From there we see that the time of mourning (which would prevent us from saying "Good Shabbat" to one another) is after Mincha, close to darkness.
All this is even more clear if we look at the other reason the Mishna Berurah (S"K 6) brings for saying Tzidkatcha by Mincha. We are saying Tzidduk HaDin (admitting the righteousness of G-d's Judgment) about the wicked people, who are going back to the torments of Gehenom when Shabbat ends, after there reprieve from the pain on Shabbat.
If so, it makes sense that the time of Judgment is close it when it gets dark and Shabbat will soon be over.