The tumor suppressor p53 is activated by stress and leads to cellular outcomes such as apoptosis and cell-cycle arrest. Its activation must be highly sensitive to ensure that cells react appropriately to damage. However, proliferating cells often encounter transient damage during normal growth, where cell-cycle arrest or apoptosis may be unfavorable. How does the p53 pathway achieve the right balance between high sensitivity and tolerance to intrinsic damage? Using quantitative time-lapse microscopy of individual human cells, we found that proliferating cells show spontaneous pulses of p53, which are triggered by an excitable mechanism during cell-cycle phases associated with intrinsic DNA damage. However, in the absence of sustained damage, posttranslational modifications keep p53 inactive, preventing it from inducing p21 expression and cell-cycle arrest. Our approach of quantifying basal dynamics in individual cells can now be used to study how other pathways in human cells achieve sensitivity in noisy environments.