With this NMR study, we show that the quality of the growth of Co/Cu multilayers depends significantly on the Co- and Cu-layer thicknesses: The initial stacking of Co layers is fcc but the hcp stacking tends to develop with increasing Co-layer thickness. On the other hand, the thicker the Cu layers are, the better the fcc growth is. By a comparison of the NMR spectra with those expected from model interface structures, we show that the interfaces between copper and thin Co layers can be described as follows. If the presence of defects in the bulk of the Co layers is excluded, the NMR spectra observed in samples with thin Co layers must be explained by an intermixing of Co and Cu over three monolayers. However, such a model does not hold true for samples with thick Co layers where the existence of bulk defects is strongly suggested. In such a case, the spectra for all the samples studied can be explained by sharp Co/Cu interfaces, as in UHV-grown multilayers, and about 0.7% impurities in the bulk of the layers or possibly columnar grains with an in-plane diameter of about 90 angstrom. Although model-agreement factors tend to favor the second model (sharp interfaces and bulk defects) rather than the first one (diffused interfaces), additional information on the nature of the bulk defects is needed.