Actually you weren't doing anything right.
Code:
Sub SplitDocumentTest()
Dim oRng As Range
Dim oDoc As Document
Set oRng = ActiveDocument.Range
With oRng
.Collapse wdCollapseStart
.EndOf Unit:=wdSection, Extend:=wdExtend
.MoveEnd wdCharacter, -1
Set oDoc = Documents.Add()
.Copy
oDoc.Content.Paste
oDoc.SaveAs2 FileName:="Specification_1", FileFormat:=wdFormatXMLDocument
Set oDoc = Nothing
.Collapse wdCollapseEnd
.MoveStart wdCharacter, 1
.EndOf Unit:=wdSection, Extend:=wdExtend
.MoveEnd wdCharacter, -1
Set oDoc = Documents.Add()
.Copy
oDoc.Content.Paste
oDoc.SaveAs2 FileName:="Specification_2", FileFormat:=wdFormatXMLDocument
Set oDoc = Nothing
.Collapse wdCollapseEnd
.MoveStart wdCharacter, 1
.EndOf Unit:=wdSection, Extend:=wdExtend
.MoveEnd wdCharacter, -1
Set oDoc = Documents.Add()
.Copy
oDoc.Content.Paste
oDoc.SaveAs2 FileName:="Specification_3", FileFormat:=wdFormatXMLDocument
Set oDoc = Nothing
End With
lbl_Exit:
Exit Sub
End Sub