the particle in 1-D box momentum and total energy(P.E + K.E (x))
As this is obviously a homework problem, I'll just give you a hint:

Observables can be measured simultaneously with arbitrarily small uncertainties if their operators commute. Compute the commutator of the momentum and particle-in-a-box Hamiltonian operators; if the commutator is zero, the answer to your question is "yes".