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In a 2001 paper, S. Montgomery-Smith provides a one-dimensional model for the three-dimensional, incompressible Navier-Stokes equations, for which he proves the blow-up of solutions associated with a class of large initial data, while the same global existence results as for the Navier-Stokes equations hold for small data. In this paper the model is adapted to the cases of two and three space dimensions, with the additional feature that the divergence-free condition is preserved. It is checked that a family of initial data constructed by Chemin and Gallagher, which is arbitrarily large yet generates a global solution to the Navier-Stokes equations in three space dimensions, actually causes blow-up for the toy model --- meaning that the precise structure of the nonlinear term is crucial to understanding the dynamics of large solutions to the Navier-Stokes equations. [ABSTRACT FROM AUTHOR]