You chose the "A" (advanced) level prefix for this thread, which would imply that you want a mathematical answer pitched at the graduate level, but the question is phrased in a nonmathematical way, so I assume the first thing we need to do is to interpret your question into more rigorous mathematical language.
One way, which is I think what DaleSpam had in mind in #2, was to interpret this as a question about a singularity. Then the answer is that singularities are a generic outcome in GR -- in some sense "most" initial conditions will lead to singularities, so the answer to your question is guaranteed to be yes.
Another possible interpretation is that you're asking about event horizons -- an event horizon is a place where time does in some loose verbal sense "stop" as "seen" by a distant observer (scare quotes because these words don't really mean anything rigorous).
The interpretation I had from reading your question is that you were asking about degeneracy of the metric. The most common situation that occurs is that we get some solution to the Einstein field equations, such as the Schwarzschild solution written in Schwarzschild coordinates, and it has a coordinate singularity in it, where the metric becomes degenerate. Since it's only a coordinate singularity, it can be removed by a better choice of coordinates. However, there is no guarantee that a metric degeneracy cannot actually happen physically. The standard formulation of GR doesn't work when the metric is degenerate, since, e.g., we always assume that we can raise and lower tensor indices at will. However, there are other formulations of GR, such as the Ashtekar formulation, that are equivalent to the standard one under normal conditions but do allow degeneracy of the metric. So I think the answer is that we don't know whether metric degeneracy is possible, and it's probably not even clear how to interpret such a possibility physically. E.g., if the metric signature becomes 0+++ or ++++ somewhere, then measurement processes presumably become impossible there, and it becomes problematic to associate the predictions of the theory with any type of observation.
