Decades of research suggest that selective attention is critical for binding the features of objects together for conscious perception. A fundamental question, however, remains unresolved: How do people perceive objects, albeit with binding errors (illusory conjunctions), when attentional resolution is poor? We used a novel technique to investigate how features are selected to create percepts of bound objects. We measured the correlation of errors (intrusions) in color and identity reports in spatial and temporal selection tasks under conditions of varying spatial or temporal uncertainty. Our findings suggest that attention selects each feature independently by randomly sampling from a probability distribution over space or time. Thus, veridical perception of bound object features arises only when attentional selection is sufficiently precise that the independently sampled features originate from a single object.