Why is the Rado–Milner paradox considered a paradox, anyway?

Question

The cardinality of $\alpha$ (where $\kappa\le\alpha<\kappa^+)$ is equal to $\kappa$ anyway. Why should it be a suprise that I can "pack" $\alpha$ into a countable number of boxes, each of a size no bigger than $\kappa^n$ ? Of course I have to use the right sequence when picking the elements from $\alpha$ when doing that (i.e. exploiting the bijection between $\alpha$ and $\kappa$).