The aim of this work is to contribute to understanding whether the partial reconnection of magnetic field lines during crashes of sawtooth oscillations in tokamaks could result from the stochasticity of the magnetic field lines. It is shown numerically that the joint action of the effects of frozen-in magnetic field lines and satellite harmonics of the magnetic perturbation can lead to widespread stochasticity; the stochasticity occupies, in particular, the near-axis region—which is a necessary condition for drops of the central temperature of the plasma during sawtooth crashes with incomplete reconnections. In spite of this, the question in the title of this work is still open. The matter is that, first, our calculations were carried out with a special choice of the perturbation and for a low central safety factor and, second, as shown in this work, previous theories investigating field line stochasticity cannot explain crashes with partial reconnections.