The dynamics of fast particles corresponding to keV in NSTX, or of alpha particles in a properly rescaled reactor-grade spherical tokamak, is computed numerically through integration of the full equations of motion. The magnetic moment μ of these particles has large oscillations, and even chaotic ones, in a sizable domain of the machine. This has both practical and physical consequences. First, when μ has large oscillations, the use of guiding-center or gyrokinetic calculations for such orbits is questionable. Second, the capability of these particles to excite Alfvénic instabilities decreases, since the velocity of the particle fluctuates with respect to the phase-velocity of the Alfvén wave, which imposes a fluctuating sign to the energy exchanges with this wave. Even when chaotic orbits are present, the conservation of the toroidal momentum sets strong constraints about the volume available to the particles, and their radial diffusion stays bounded.