We have studied the rotational mobility of a mixture of butane and pentane on zeolite ZK-5 as a function of temperature from H-1-C-13 cross-relaxation at 4.7 and 11.7 T. In the studied temperature range (130-320 K), the sorbed alkanes are practically confined inside the alpha and gamma cages of ZK-5. This is confirmed by well resolved peaks in magic-angle-spinning (MAS) C-13 NMR spectra for each alkane in each cage. Despite the translational confinement, the observed NMR relaxation for the methylene groups of butane and pentane reflects extensive rotational motion inside cages. The maximum cross-relaxation rates indicate a combination of ultrafast rotation about the alkane axis (including internal bond rotations) and a slower tumbling of the molecular axis itself. Two temperature regimes can be distinguished. Above 200 K, the effective tumbling times derived from the initial cross-relaxation show Arrhenius behavior with activation energies between 5 and 15 kJ/mol. Below 200 K, the motion becomes less activated. We consider various possible explanations and discuss three models: (A) a combination of activated and nonactivated motion, (B) a Gaussian distribution of activation energies, (C) concerted sorbate-zeolite motions. Model C involves a flexible zeolite lattice, which accommodates the binding of molecules by small dynamic deformations.