  We show that in {\em ballistic} mesoscopic SNS junctions the period of
critical current vs. magnetic flux dependence (magnetic interference pattern),
$I_c(\Phi)$, changes {\em continuously and non-monotonically} from $\Phi_0$ to
$2\Phi_0$ as the length-to-width ratio of the junction grows, or temperature
drops. In {\em diffusive} mesoscopic junctions the change is even more drastic,
with the first zero of $I_c(\Phi)$ appearing at $3\Phi_0$.
  The effect is a manifestation of nonlocal relation between the supercurrent
density and superfluid velocity in the normal part of the system, with the
characteristic scale $\xi_T = \hbar v_F/2\pi k_BT$ (ballistic limit) or
$\tilde{\xi}_T = \sqrt{\hbar D/2\pi k_BT}$ (diffusive limit), the normal metal
coherence length, and arises due to restriction of the quasiparticle phase
space near the lateral boundaries of the junction. It explains the
$2\Phi_0$-periodicity recently observed by Heida et al. (Phys. Rev. B {\bf 57},
R5618 (1998)). We obtained explicit analytical expressions for the magnetic
interference pattern for a junction with an arbitrary length-to-width ratio.
Experiments are proposed to directly observe the $\Phi_0\to 2\Phi_0$- and
$\Phi_0\to 3\Phi_0$-transitions.
