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: 
You are simply looking at it from an observer's viewpoint. Yes, looking from outside, matter tends to asymptotically approach but never reach the event horizon. If you were part of that matter spiraling into a black hole, there would be no problem reaching the horizon, crossing it, and going right down to the singularity. The event horizon is not a physical barrier.
You could be free falling, and your time would not be infinitely dilated. So the answer is yes they can form easily in a finite time.
Ben's answer did not show that black holes/event horizons can ever form from a distant observer's (e.g. our) perspective. In our frame of reference, strictly according to GR, no event horizon could ever have formed. For us, it doesn't matter what happens in the reference frame of collapsing matter of a star that went super nova, no matter how many Penrose diagrams one shows.