I made a cylinder and removed the top face, so I made a bucket, but when my character jumps inside, I fall right through. The bucket is huge. It has to be. Adding collision bounds means I can't go in the bucket at all. I also tried putting collision bounds on my character, but still failed.
I haven't used the BGE for a decade, but I reckon physics on meshes without depth aren't the best.
If your bucket is literally a cube minus a face, then you can add and apply a Solidify Modifier to give it some depth.
Also, make sure you applied that object's scale with ⎈ CtrlA.
Then, also check that object's physic settings:
Collisions are set to "Convex Hull" by default if I remember correctly, it will not work for concave shapes like a bucket, you need to set it to Mesh instead.
There might be other settings to play with, like the margins, damping, ... Though by then it should just tweak how the impact looks like, not whether or not it actually works.
