We recently put a new ceiling in our family room and put in 6" faced fiberglas batts. Question - we have left over batts. Can they be installed on top of the existing batts?

Only if you take the facing off. You can lay as many layers of fiberglass batts on top of each other as you want and can even mix batts with blown in insulation. What you can't do is have more than one vapor barrier as this will tend to trap moisture between those layers.

If you have access from the top, then easy enough to peel the facing off and add the extra fiberglass on top of what is already there.

6" is not a whole lot of insulation in an attic.
When do you live?

http://www.energystar.gov/inde...ent_insulation_table
Also make sure you do not install it where you will be blocking the soffit vents.
Your were also suppost to air seal all the air leaks where any plumbing, wiring, lighting or ceiling fans where installed through the top plates or ceilings.