High-throughput imaging has led to an explosion of observations about cell-size homeostasis across the kingdoms of life. Among bacteria, "adder" behavior-in which a constant size increment appears to be added during each cell cycle-is ubiquitous, while various eukaryotes show other size-homeostasis behaviors. Since interactions between cell-cycle progression and growth ultimately determine such behaviors, we developed a general model of cell-cycle regulation. Our analyses reveal a range of scenarios that are plausible but fail to regulate cell size, indicating that mechanisms of cell-cycle regulation are stringently limited by size-control requirements, and possibly why certain cell-cycle features are strongly conserved. Cell-cycle features can play unintuitive roles in altering size-homeostasis behaviors: noisy regulator production can enhance adder behavior, while Whi5-like inhibitor dilutors respond sensitively to perturbations to G2/M control and noisy G1/S checkpoints. Our model thus provides holistic insights into the mechanistic implications of size-homeostasis experimental measurements.