Quake3World

I've noticed in some PC and console games that there is screen tearing even at low FPS when the FPS is much lower than the refresh rate of the monitor. Isn't that supposed to happen only at very high FPS when frames overlap?

yes, its when its out of sync
for consoles, set it to the monitor's native res or get a better screen if the game's in 1080p

You notice it more at lower FPS because the redraw represents a larger degree of change from the previous frame.

When you have high FPS and the screen redraws, the new frame is visually closer to the previous frame.