We discuss the constraints that a COBE-DMR-like experiment can set on the CMB (Cosmic Microwave Background) quadrupole anisotropy, once that Galactic emission and receiver noise are taken into account. We use a simple model for describing the diffuse emission of our Galaxy at millimeter wavelengths, and we estimate that the minimum Galactic quadrupole occurs at lambda approximately 3-4 mm, with an amplitude almost-equal-to 50-mu-K. Multifrequency experiments, such as COBE-DMR, can easily disentangle the Galactic and CMB quadrupoles. We discuss several methods for isolating the CMB quadrupole: its measurement, according to our simulations, turns out to be limited in sensitivity mostly by detector noise and much less by Galactic contamination. At the expected level of sensitivity, COBE should be able to set a very conservative upper limit to the rms CMB quadrupole of congruent-to 80-mu-K. This upper limit would be fatal for open (OMEGA = 0.1 - 0.4) pure baryonic, fully ionized universes, with a primordial spectral index n = -1. In flat, biased (b congruent-to 1.5), cold dark matter models, a fraction less-than-or-similar-to 50% of the cosmic observers should detect a CMB quadrupole anisotropy.