Question: One thing I know about black holes is that an object gets closer to the event horizon, gravitation time dilation make it move more slower from an outside perspective, so that it looks like it take an infinite amount of time for the object to reach the event horizon. It seems like a similar process should slow the formation of the black hole itself: As the star collapses, its gravitational time dilation make itself collapse more slowly. This make me wonder, are what astronomers claim to be black holes really black holes, or are they stars that progressively make themselves more similar to one without actually reaching the stage of having an event horizon?

EDIT: Contemplating one answer, I realize the question is ambiguous. What does finite time mean in general relativity. Here is a less ambiguous question: Is there a connected solution of 3+1 dimensional general relativity with one space-like slice not have a singularity, and another space-like slice having one.

Answer: 
To begin with, there is a connected solution of 3+1 GR in which particles fall to the singularity in finite time. In particular, Gullstrand-Painleve coordinates do this. The big difference with Schwarzschild coordinates is that the speed of light depends on direction: light moves into a black hole faster than it moves out. See:  

<http://en.wikipedia.org/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates>

For the formation of a black hole in these coordinates, see: 

Phys.Rev.D79:101503,2009, J. Ziprick, G. Kunstatter, *Spherically Symmetric Black Hole Formation in Painlevé-Gullstrand Coordinates*  

<http://arxiv.org/abs/0812.0993>

For the generalization of Gullstrand-Painleve coordinates to the rotating black hole, see the very readable paper that gives an intuitive explanation for what is going on, see:  

Am.J.Phys.76:519-532,2008, Andrew J. S. Hamilton, Jason P. Lisle, *The river model of black holes*
<http://arxiv.org/abs/gr-qc/0411060>

Note: that the above paper is peer reviewed and shows that yes indeed, particles falling past the event horizon travel with speeds greater than the 1 (in GP coordinates). In GR, the speeds of objects depend on the choice of coordinates. Consequently, this exceeding of the speed 1 is not equivalent to exceeding the speed of light. In GP coordinates, a light beam moving towards the singularity inside the event horizon also moves at speed greater than 1. Consequently, there is no violation of special relativity.
It seems like, if there is a sufficient amount of matter localized during the Big Bang, then a Black Hole could form. But after that, a massive star, as seen by the rest of the universe could never shrink to the size of the event horizon.

In other words, a Black Hole would have to preexist in the universe to be an actual Black Hole.