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: 
(this answer addresses the new question)

As a consequence of the singularity theorems, it is not only possible but (arguably) inevitable for singularities to form in a finite amount of "time" in a physically reasonable spacetime. The word "time" in this context means "proper time along a specific timelike geodesic". For example, if there is a trapped surface\* in spacetime, then a singularity will appear within a finite amount of proper time (along a timelike geodesic) in the future of that surface; so, an observer sitting in a collapsing star will reach the singularity in finite time. Thus, the collapse of matter is one possible way to create a singularity "out of nothing". If your spacetime is globally hyperbolic and you foliate it by Cauchy surfaces you can say in a much more "universal" way that the singularity didn't exist at time [$t\_{0}$] and came to exist at time [$t\_{1}$]. 

I should point out that the singularities are a generic feature of physically reasonable spacetimes; take a look at the [Hawking-Penrose theorem](https://en.m.wikipedia.org/wiki/Penrose%E2%80%93Hawking_singularity_theorems)- it applies in very general situations.

Also, as the original question was about black holes and not singularities, I should advise you to make a clear distinction between the two concepts. Trapped surfaces form due to the condensation of matter (this is the famous Schoen-Yau theorem), and under a certain extra hypothesis, these surfaces will be hidden inside black holes. This extra hypothesis is the well-known (weak) Cosmic Censorship Conjecture (CCC). If it does not hold, gravitational collapse can create naked singularities, that is, singularities not "causally hidden" by the event horizon of a black hole. Much of what is known in general about black holes depend crucially on the CCC.

\*A trapped surface is a two-dimensional spacelike compact surface such that the null geodesics departing from it are accelerating towards each other - mathematically, we say that the expansion of the congruence of future-directed null geodesics orthogonal to the surface is negative.
I wouldn't know that any of the answers above has shown that, from an outsider's perspective, anything can ever reach the horizon, which was essentially the question of the OP. From an in-falling observer's point of view, there is no problem because kinematic time dilation and gravitational time contraction of the rest of the universe ("looking" in the rear view mirror of an in-falling observer's space ship, let's say) cancel each other out, exactly. But from an outsider's point of view, this is never the case.

Btw, that does not contradict any of the phenomena of black holes like jets and accretion disk-dependent events.

P.S. I suggest to not first think about e.g. light signals when we want to understand the nature of relativistic phenomena. The reality can always be "retro-engineered" from light signals, their arrival time and location and their redshifts, but that's very complicated and confusing to begin with. In order to understand things, we should first assume an instant information transfer.