We study the (2+1)-dimensional Gross-Neveu model at non-zero chemical potential and subjected to a homogeneous background magnetic field. We do so both analytically, in the limit of an infinite number of fermion flavors in which mean-field approaches become exact, as well as on the lattice for a single flavor. The rich and exotic phase structure observed in the mean-field limit is found to be destroyed when strong quantum fluctuations are present in the system. Instead, in the phase of spontaneously broken chiral symmetry the magnetic field enhances this breaking for all choices of parameters. As a byproduct, we find indications for a first-order phase transition in the chemical potential for vanishing magnetic field but also provide hints that this could rather be a finite-size than a finite-flavor-number effect.