If you insert a row, the Change event will fire, and your change event reprotects the sheet. You'll need to disable events while inserting:
Const strPASSWORD As String = "gideon"
Sub InsertARow()
'
' InsertARow Macro
' Macro recorded 01/09/2008 by v14dmin
'
' Keyboard Shortcut: Ctrl+i

If ActiveSheet.Name <> "Timetabled Service" Then Exit Sub

ActiveSheet.Unprotect strPASSWORD
Worksheets("Frequent Calculate").Unprotect strPASSWORD
on error resume next
application.enableevents = false
ActiveCell.EntireRow.Insert
ActiveCell.End(xlToLeft).Select
Range("Master_Row").Copy Destination:=ActiveCell
application.enableevents = True
on error goto 0

Worksheets("Frequent Calculate").Protect strPASSWORD 
ActiveSheet.Protect strPASSWORD, DrawingObjects:=True, Contents:=True, Scenarios:=True _
        , AllowInsertingRows:=True, AllowDeletingRows:=True, AllowSorting:=True

End Sub