While writing an answer to this question, I started doubting about the interpretation of the uncertainty principle for the particle in a box.
In the 1-dimensional particle in a box problem, explicit solutions for the energy eigenstates exist, and are essentially of the form $\sin nx$ outside the support of the potential, and 0 where the potential is infinite. These somehow feel as if they should have definite momentum (up to direction), since they are free where the potential is 0, so that their energy is kinetic energy, and they cannot be in the region where they are not free.
On the other hand, position is strictly restricted to some interval. Consequently, any state must have bounded support in positional representation, and one would say there can be no states with bounded support in momentum space (by Paley-Wiener if you want). Indeed, the energy eigenstates involve an infinite number of momentum eigenstates (as they are zero outside an interval).
What is the way out? I think it must be that part of the energy in the energy eigenstates is potential energy. To see that, we can think of the idealized infinite potential as a limit of some sequence of finite potentials, which also is necessary in order to be able to give a meaning to the second derivative appearing in the Hamiltonian. This second derivative in the boundaries of the well tends to a multiple of a Dirac mass which has to be compensated in a very essential way by the term $V(x)\psi(x)$ in the Schrӧdinger equation $\psi''(x) + V(x)\psi(x) = \psi(x)$ (up to constants), so that in any approximation to the infinite potential will have a non-negligible contribution from the potential energy.
Is this a correct interpretation?