Three mathematicians named Courant, Friedrichs, and Lewy created a criterion that, if violated, would lead to the "blowing up" of a finite-difference weather prediction model. This CFL criterion is:

Because of the CFL criterion, a modeler cannot arbitrarily choose a horizontal grid spacing without also taking into account the time step of the model. If you want fine horizontal resolution to see small-scale weather, you must have fine time resolution too. Otherwise, the model "blows up"

What does it look like when a model "blows up"? Try it yourself using the below applet. Troughs and ridges turn disintegrate when the horizontal grid spacing and time step don't satisfy the CFL criterion. The forecast is ruined. (See the detailed instructions, below.)