Why did you have to download the Avery J8651 template? It's already in my Word 2003 label list.
Anyway - do the 'All borders' printing the test sheet on plain A4 so that it can be held up to the light to compare with the cut lines on label sheet. (It helps if you write on the paper which is top and bottom)
That should show you which borders are not lining up right with the cut lines.
(When printing proper label sheet change to 'No Borders')