One of the most important questions in the physics of gravitation phenomena is whether gravitational collapse can lead to the formation of singularities which are not hidden by an event horizon. The Cosmic Censorship Conjecture (CCC) represents the hope that such a drastic event cannot happen in realistic physical situations. However, in the recent past several counter examples to the CCC were demonstrated by several researchers in situations of spherically symmetric gravitational collapse. The disturbing aspect about these counter examples is that they are strong naked singularities—they can crush matter to zero volume and can have a disastrous influence on causal physics. We examine these counter examples for their physical content by working through the dynamical collapse of inhomogeneous dust and argue that these are not physically acceptable counter examples. Our main result is that the singularities when naked are weak and when strong, strongly censored. The strong naked singularities in the counter examples do not arise from dynamical collapse; they result from the intrinsically singular nature of the initial density distributions chosen. The CCC seems to remain robust as far as spherically symmetric collapse is concerned.