In this article, we describe the characterization of a newly fabricated amorphous alloy system (TiZrNbCu)1−xCox covering a broad composition range from high-entropy (HEA) to Co-rich alloys (x ≤ 0.43). We investigated thermal stability, atomic and electronic structure, and magnetic and mechanical properties as a function of chemical composition x. One of the important findings is that all studied properties change their dependence on concentration x within the HEA range. In particular, it has been found that the average atomic volume deviates from Vegard’s law for x > 0.2, the concentration for which the average atomic packing fraction suddenly changes. The valence band structure, studied with ultraviolet photoemission spectroscopy, shows a split-band shape with 3d-states of Co approaching the Fermi level on increasing x. Due to the onset of magnetic correlations, magnetic susceptibility rapidly increases for x > 0.25. Very high microhardness increases rapidly with x. The results are compared with those for similar binary and quinary metallic glasses and with those for Cantor type of crystalline alloys.