At the beginning of the code I blow away any chart objects I find with

Dim co As ChartObject
For Each co In mySheet.ChartObjects
co.Delete
Next

And for good measure (this is probably all you have to do)

mySheet.ChartObjects.Delete

Afterwards, Chart.ChartObjects.Count is == 0

Later, I review the charts on the sheet with

For Each co In ws.ChartObjects
sTitle = co.Chart.Name
MsgBox ("has the name " & sTitle)
Next

Only one chart is found, but it's name is incrementing. This leads me
to believe that I really didn't blow away everything and that I can't
be sure I'm starting with a clean slate.

If I delete the sheet and recreate it, the chart names reset.

Why does this name persist?