Whether 6 bins will fit depends on the size and shape of the bins. My D uses the Sterlite modular stacking drawers under her bed and they’ve worked wonderfully to increase storage. They’ve also been useful for packing up at year end. She raises her bed to a height of @ 2 ft. and uses 3 large drawers as the base (@24" wide by @ 10 1/2 " high) then tops those with 4 medium (12" wide) drawers and 3 small drawers (8" wide). They fit neatly under her bed and hold bulky clothes, first aid and cleaning supplies, extra sheets, etc.
I was certain that her stuff wouldn’t fit into her 1/2 of the dorm room when she moved in freshman year, but I was wrong. Once the bed was made up, desk supplies and clothes unpacked and put away, posters and decorations hung, etc. that mass of stuff was seamlessly absorbed by the room.
Draw the room to scale and plan where the things your D is bringing will go to see whether everything will fit. You’re likely to have a harder time fitting things into the car (space bags are great for shrinking bulky bedding item) than the dorm.