A graded river conveys its sediment load without net aggradation or degradation. Grade is thought to represent the equilibrium state of a river system subject to steady allogenic forcing. We address the concept of grade in linked depositional systems through a combination of mathematical modeling and flume experiments. We develop a moving boundary morphodynamic model of fluviodeltaic sedimentation and use it to quantify the conditions necessary for a state of sustained grade in an alluvial-deltaic system developing on a uniformly sloping shelf. To test our theory, we conducted a series of flume experiments in which we constructed laboratory-scale fluviodeltaic systems under specific relative sea level histories. On large spatiotemporal scales, our theoretical and experimental results suggest an alternative view of grade as an intrinsically nonequilibrium state that requires a fall in relative sea level. Under conditions of stillstand or rise in relative sea level, the equilibrium (graded) profile does not exist, and the alluvial river will always aggrade. With a steady sediment supply, alluvial grade can be sustained if relative sea level falls with a specific, square-root-of-time dependence. Common sequence-stratigraphic conceptualizations of alluvial river response to sea level change suggest that rivers aggrade and degrade in response to relative sea level rise and fall, respectively. We show that rivers can remain aggradational if the rate of sea level fall is sufficiently small. The boundary between aggradational and degradational states is the graded state described here: that associated with a relative sea level fall that varies with the square root of time.