Can a child get chicken pox even though they have had the varicella vaccine?

Yes: Although the vaccine is the right thing to do, a percentage of children receiving the vaccine find that it does not confer immunity on everyone. But the more people that are immunized, the fewer people have it to spread.Thus "community immunity" helps also in protection.

Yes: This vaccine is a modified strain of the wild varicella virus ; is thought to confer lifelong immunity in 85 -90 % after 1 dose. A second dose is recommended for all kids and adults, to pick up the stragglers with poor initial response and brings up the protection rate above 95%.One dose confers protection from the major complications.