If you could get up high enough in the sky, then you'd see that some rainbows continue below the horizon. That's because when the sun and rain combine to make a rainbow, they really make a full-circle rainbow. We can't see all of the circle, because the horizon blocks it from our view. Pilots high in the sky do sometimes report seeing genuine full-circle rainbows.
The only way to see the full circle of a rainbow in the sky is to be above the raindrops and have the sun behind you.