We investigated decays of ^{51,52,53}K at the ISOLDE Decay Station at CERN in order to understand the mechanism of the β-delayed neutron-emission (βn) process. The experiment quantified neutron and γ-ray emission paths for each precursor. We used this information to test the hypothesis, first formulated by Bohr in 1939, that neutrons in the βn process originate from the structureless "compound nucleus." The data are consistent with this postulate for most of the observed decay paths. The agreement, however, is surprising because the compound-nucleus stage should not be achieved in the studied β decay due to insufficient excitation energy and level densities in the neutron emitter. In the ^{53}K βn decay, we found a preferential population of the first excited state in ^{52}Ca that contradicted Bohr's hypothesis. The latter was interpreted as evidence for direct neutron emission sensitive to the structure of the neutron-unbound state. We propose that the observed nonstatistical neutron emission proceeds through the coupling with nearby doorway states that have large neutron-emission probabilities. The appearance of "compound-nucleus" decay is caused by the aggregated small contributions of multiple doorway states at higher excitation energy.