It is not clear from your code exactly what you are trying to do. There could be a few errors in it as it stands:
1) The first element in oneList will be element 0, so I would expect to see:
Should listItem and list_Item be the same object?
You should Dim the listItem to an appropriate type each time you assign it, otherwise it will just change the pointer for all added elements.
If so your code should be more like this:
For y=0 To 4
Dim listItem As String = oneList.Get(y)
anotherList.Add(listItem)
Next