We combined experimental investigations and theoretical calculations to unveil an abnormal magnetic behavior caused by addition of the nonmagnetic element Cu in face-centered-cubic FeNiCoMn-based high-entropy alloys (HEAs). Upon Cu addition, the probed HEAs show an increase of both Curie temperature and saturation magnetization in as-cast and homogenized states. Specifically, the saturation magnetization of the as-cast HEAs at room temperature increases by 77% and 177% at a Cu content of 11 and 20 at. %, respectively, compared to the as-cast equiatomic FeNiCoMn HEA without Cu. The increase in saturation magnetization of the as-cast HEAs is associated with the formation of an Fe-Co rich phase in the dendritic regions. For the homogenized HEAs, the magnetic state at room temperature transforms from paramagnetism to ferromagnetism after 20 at. % Cu addition. The increase of the saturation magnetization and Curie temperature cannot be adequately explained by the formation of Cu enriched zones according to atom probe tomography analysis. Ab initio calculations suggest Cu plays a pivotal role in the stabilization of a ferromagnetic ordering of Fe, and reveal an increase of the Curie temperature caused by Cu addition which agrees well with the experimental results. The underlying mechanism behind this phenomenon lies in a combined change in unit-cell volume and chemical composition and the related energetic stabilization of the magnetic ordering upon Cu alloying as revealed by theoretical calculations. Thus, the work unveils the mechanisms responsible for the Cu effect on the magnetic properties of FeNiCoMn HEAs, and suggests that nonmagnetic elements are also crucial to tune and improve magnetic properties of HEAs.