Another non-optimal method is to use the selection object. It has the benefit of working across page breaks.
Code:
Sub HowManyLines()
Dim aRng As Range, i As Integer, aTbl As Table
Set aTbl = ActiveDocument.Tables(1)
Set aRng = aTbl.Cell(1, 2).Range
aRng.Select
Selection.Collapse Direction:=wdCollapseStart
Do While Selection.Range.End < aRng.End
i = i + 1
Selection.MoveDown Unit:=wdLine, Count:=1
Loop
MsgBox "Cell line count: " & i
End Sub
The problem with this method is that it relies on moving the selection. I wasn't able to find a Range.MoveDown option but if one existed then it would be better than using Selection.