Through a simple illustrative example, it is shown that instantaneous convective adjustment schemes, of the type used in general circulation models to parametrize nonhydrostatic convective processes, lead to the spontaneous emergence of the smallest resolved horizontal scale: the grid mode is unstable regardless of the strength of the horizontal diffusivity. Convective adjustment vertically mixes properties at each grid-point, irrespective of the horizontal distribution of such properties. Thus, horizontal spatial gradients are amplified by convective adjustment, as long as adjustment is faster than the horizontal diffusion (or advection) time between neighboring grid-points. In the example presented here, the grid-scale instability is a global attractor and can only be “suppressed” by inaccurate time-stepping, or by the finite computational representation of numbers. This clarifies that the “grid-mode” is not a computational instability, but an intrinsic property of instantaneous convective adjustment schemes. A smooth solution, without grid-scale gradients, also exists, but it is unstable to infinitesimal perturbations for all values of the external parameters. We emphasize that the spatial average of the grid-mode differs substantially from the spatial average of the smooth (but unstable) solution.