Subject: What determines if two species can produce a viable fertile offspring?

I was talking to a student about a show in which an alien and a human naturally 
had a baby. This student explained that there was a disagreement about how 
genetically and thus biologically similar the two races would have to be to 
produce an offspring. Despite its source, I found the question to be very 
enlightened and attempted to find an answer.

At first, I thought that the number of chromosomes would limit interspecies 
breeding, but a question here about genetic mutation affecting breeding 
disproved that theory. I found a report on interspecies breeding of monkeys but 
it didn’t cover genetics. I have tried looking into interspecies breeding 
between Homo erectus and Homo sapiens, but that information is sparse and 
limited in its work and conclusions. The only thing that they all seem to agree 
on is that there has never been any Neanderthal mtDNA found in modern humans. 
Thus it seems that Neanderthal and Cro-Magnon were incapable of producing 
viable offspring.

Therefore, I was hoping that you could help me understand what limits 
interspecies and/or intergeneric breeding? Why can some species interbreed, 
despite vast chromosomal numbers, and others can’t? What in the genetic makeup 
dictates the ability to produce viable fertile offspring? How similar 
genetically would two creatures have to be to reproduce those offspring?

Re: What determines if two species can produce a viable fertile offspring?