The Schwarzschild interior solution is a perfectly valid solution of Einstein's equations. Certain of the assumptions it makes are unlikely to be true for an actual physical collapse, however.
One of the big issues is that the Schwarzschild solution is perfectly spherically symmetrical. A real dust cloud will never be symmetrical. Suppose it has clumps - the question is, will those clumps tend to spread out over time, through diffusion, or will they tend to grow, because of gravity attracting more material to the clump - and hence grow?
My understanding of the situation is that in the interior, gravity is so intense that the effect of gravity is likely overwhelm any mechanism that could make the clump to "spread itself out". Therefore, the assumption of perfect spherical symmetry is one of the suspect assumptions. The Schwarzschild solution is not "stable", a small departure from the perfect symmetry will tend to grow larger.
People who have looked at the issue (specifically, Kip Thorne) seem to prefer the BKL solution for the non-rotating case.
http://en.wikipedia.org/w/index.php?title=BKL_singularity&oldid=431041130.
This is only one of the issues. The other issue is that any actual solution will probably have some angular momentum. While we do have a rotating spherical symmetrial solution, the Kerr solution, it again makes the assumption that the geometry is perfectly symmetrical. (It's also not the Schwarzschild solution which we started out discussing, so it's another reason the Schwarzschild solution won't likely be found in reality.). Again, this assumption of perfect symmetry is suspect. The seemingly innocuous assumption of spherical symmetry leads to some unpleasant and rather unphysical "artifacts" in the Kerr solution, such as infinite blueshifts at the "inner horizon". "Mass inflation" see
http://arxiv.org/abs/0811.1926, is expected to eliminate such infinite blueshifts, and to very significantly alter the nature of the "inner horizon".