Can two points be added?

The reason I ask is because when I think about it all I see is vector addition. I understand the difference between vectors and points. I know we used to talk about points on the number line as a kid, and we added numbers, but I can't assume equivalency.

It also feels strange to say "if I add this location with this other location I get a new location."

What is the answer, and what is the root of my confusion?