This paper considers the motion of charged particles in irregular, statistically homogeneous, two-dimensional magnetic fields, a situation which might arise in multiple reconnecting current sheets and in two-dimensional MHD turbulence. The (test) particle dynamics is investigated with respect to the existence of a kinetic energy threshold, beyond which the trajectories become unbounded and effective diffusion occurs. Reformulating the Lorentz force kinematics in terms of an effective potential, the threshold problem reduces to a two-dimensional continuum percolation problem. Assuming that any realizations of Ay(x, z) and -Ay(x, z) appear with equal probability, a rigorous condition for unbounded trajectories in terms of the motion invariants is derived: $ mathcal{E} > frac{1}{2}P_y^2 $, with $ mathcal{E} $ the kinetic energy and Py the canonical momentum in the invariant direction. Its validity is verified by means of Monte-Carlo simulations, and the effect of an additional electric field on particle acceleration is discussed. The percolation criterion allows conclusions about the mobility and origin of energetic particles in turbulent regions of the magnetospheric tail plasma sheet.