http://en.wikipedia.org/wiki/Close-in_weapon_system
The CIWS is only used if the SeaSparrow missiles (which have a 40+ mile range) and RAMs do not hit their target.
http://en.wikipedia.org/wiki/ESSM
http://en.wikipedia.org/wiki/RIM-116_Rolling_Airframe_Missil...
Those would only be used if the jamming equipment/planes failed.
Don't forget that carriers always have several ships traveling with them with even more anti-missile firepower.
It would literally take a nuclear missile, severe incompetence, or numerous malfunctions on multiple well-tested systems to hit an aircraft carrier with a missile. Plus balls, the US will come after you hard if you mess with one of it's carriers.
But yeah, shooting a missile with multiple nuclear warheads is about the only thing that can be done to take out a carrier, and that's a pretty awesome claim.
Oh, I guess a railgun would work too, those would be the real carrier killers. The US DoD should definitely assign some extra IT security personnel to the contractors working on railguns.