Medium and larger dry flies can get smushed in the Anglers Image thin boxes. I only recommend them for 20 and smaller dries, maybe some 18's. I mostly use them for nymphs and midge dries.
I don't have the problem with spinners though, they are pretty flat. I use synthetic fiber spinner wings, not feathers, so YMMV.
There's about 1/3" clearance between magnet and lid in the Anglers Image boxes. The foam ones have a bit less clearance, but the hook goes down into the foam insert. That helps with parachutes, but collar hackles get compressed. There are some slightly thicker magnetic boxes, but I've only seen a single compartment magnetic model.
As I tie, I often put the flies into a small plastic container that comes with Dai Riki hooks. They have more inside clearance than the Anglers Image boxes, about 1/2". I find this small difference enough to safely store dry flies up to about a 14.