Intermolecular multiple-quantum coherences between bulk water and a glycoprotein fragment at modest concentration (20 mM) have been experimentally produced and detected, although such coherences are inconceivable in the normal theoretical framework of nuclear magnetic resonance. A density matrix treatment explains these results by including the long-range dipolar interaction between spins and by discarding the high-temperature approximation. These results imply that peak intensities (critical for structural determinations) can be distorted in many gradient experiments, and show that magic-angle gradients provide substantial improvements with reduced gradient strengths. They also suggest methods for contrast enhancement in magnetic resonance imaging.