High-entropy alloys (HEAs) attract much attention as possible radiation-resistant materials due to their several unique properties. In this work, the generation and evolution of the radiation damage response of an FeNiCrCoCu HEA and bulk Ni in the early stages were explored using molecular dynamics (MD). The design, concerned with investigating the irradiation tolerance of the FeNiCrCoCu HEA, encompassed the following: (1) The FeNiCrCoCu HEA structure was obtained through a hybrid method that combined Monte Carlo (MC) and MD vs. the random distribution of atoms. (2) Displacement cascades caused by different primary knock-on atom (PKA) energy levels (500 to 5000 eV) of the FeNiCrCoCu HEA vs. bulk Ni were simulated. There was almost no element segregation in bulk FeNiCrCoCu obtained with the MD/MC method by analyzing the Warren–Cowley short-range order (SRO) parameters. In this case, the atom distribution was similar to the random structure that was selected as a substrate to conduct the damage cascade process. A mass of defects (interstitials and vacancies) was generated primarily by PKA departure. The number of adatoms grew, which slightly roughened the surface, and the defects were distributed deeper as the PKA energy increased for both pure Ni and the FeNiCrCoCu HEA. At the time of thermal spike, one fascinating phenomenon occurred where the number of Frenkel pairs for HEA was more than that for pure Ni. However, we obtained the opposite result, that fewer Frenkel pairs survived in the HEA than in pure Ni in the final state of the damage cascade. The number and size of defect clusters grew with increasing PKA energy levels for both materials. Defects were suppressed in the HEA; that is to say, defects were “cowards”, behaving in an introverted manner according to the anthropomorphic rhetorical method.