  The ``cosmic censorship conjecture'' asserts that all singularities arising
from gravitational collapse are hidden within black holes. We investigate this
conjecture in a setup of interest for tests of General Relativity: black hole
solutions which are parametrically small deviations away from the Kerr
solution. These solutions have an upper bound on rotation, beyond which a naked
singularity is visible to outside observers. We study whether these (generic)
spacetimes can be spun-up past extremality with point particles or accretion
disks. Our results show that cosmic censorship is preserved for generic
parameterizations. We also present examples of special geometries which can be
spun-up past extremality.
