No, this is a common misunderstanding. Neanderthals did live at the same time as our ancestors did and they also shared a common ancestry with modern humans however, they are not an ancestor of modern humans. They are in fact a different species.
Common ancestors are the ancestors you and a relative of yours have in common. For example, your first cousin and you have the same grandparents. The grandparents are your and your first cousins common ancestors.