I hovered the mouse cursor over NewFileName and noticed a little square after the text. This square does not appear in the Immediate Window when you use Debug.Print NewFileName or even MsgBox NewFileName.
I got rid of the annoying little square with two string functions:
Dim TextBoxText As String
TextBoxText = ActiveDocument.Shapes("Text Box 3").TextFrame.TextRange.Text
NewFileName = Left(TextBoxText, Len(TextBoxText) - 1)
When I stepped through the code again and hovered the mouse cursor over the NewFileName variable, the little square was gone and the .SaveAs step executed like a dream.