As an added note, I created a new label with Avery 5164 and nothing on the label other than the shipping info. No return address or logo and I still had the same problem. Only 4 labels show up in preview mode.
For a given printer, there's a maximum printable area on a given letter sized sheet of paper. It can only print so close to the edges. This differs with each printer. If you add up the header + footer + 6 times the height of the body layout part, your layout is at least a pixel or two taller than this printable area and thus FileMaker prints the last row of labels at the top of the next page.
Normally, very small 1 or 2 pixel reductions in a layout part's height is enough to get all label rows to print without messing up the alignment of printed output to label. You can select a layout part while in layout mode by clicking the part label. Then you can click the units next to one of the Position boxes in the inspector until you see points or pixels. Then you can make a very small change in the layout part height followed by previewing your layout to see if the change was enough to get back that bottom row of labels.