For almost every flight I've been on (I fly US), the GAs announce that smaller items must go under the seat AND everyone should hold onto their coats until the overhead bins are filled. Personally, I hold onto my coat... I'm not big into having my expensive coat smashed under someones dirty rollerboard wheels.

That being said, I would never dare to remove someones item from the bin, even if it looked like it could fit under their seat. Maybe I have too much faith in humanity, but I would just assume that they had a valid reason for doing so.