The incoming Vaidya spacetimes are perhaps the simplest toy models for the dynamical formation of a Schwarzschild black hole from nonsingular initial conditions, arising from the collapse of a spherically symmetric cloud of null dust. I'll discuss the structure of these spacetimes and demonstrate, via the analysis of a simple ODE, that they can also exhibit the formation of naked singularities, singularities visible from infinity, accompanying the black hole provided only that the null dust accumulates slowly. More pointedly, I'll demonstrate that there is a sense in which the emergence of naked singularities is generic within this class of spacetimes. This observation is of note for the weak cosmic censorship conjecture, a fundamental postulate as to the nature and extent of solutions to the Einstein equations when coupled to "reasonable" matter.