Could you settle an argument?  Can  a lion reproduce with a tabby cat?  A rattlesnake with an anaconda?  Is the only issue controlling the possibility of reproduction whether or not the two animals have the same number of chromosomes?  I thought that two animals had to be of the same genus, at least.

Hi, Snowy.  Others who are more knowledgeable will have to chip in, but I can tell you a couple of things.  First of all, lions can and do reproduce with tigers: the result of this cross-breeding is called either a "liger" (if the lion is male and the tiger is female) or a "tigon" (vice versa).  Ligers are particularly interesting because they are huge: by some distance the biggest cats in the world, twice the weight of a male lion.  You can get an idea of liger size from a series of photographs, judged legitimate by snopes.com:
http://www.snopes.com/photos/animals/liger.asp
and there is a (rather cheesy) video here:
http://www.maniacworld.com/liger.html

Obvious physical barriers would prevent a lion and a tabby cat from mating, but whether offspring could be produced in a lab I don't know.  Others might.

Do two animals have to be in the same "genus" in order to be able to produce offspring together?  Ah, can open, worms everywhere!  First, what is a "genus"?  Probably no two taxonomists in the world have ever been in full agreement on this, and may genera have been split, relumped, reresplit and so on to the point where one is tempted to throw ones hands up and resign.  For example, the blue whale is variously known as Balaenoptera musculus and Sibbaldus musculus, according to the beliefs and preference of individual workers. Similarly, the American black bear may be Ursus americanus or Euarctos americanus, and the lion Felis leo or Panthera leo or even its own genus, Leo leo.

So there is no rigorous way of determining whether or not two animals belong in the same genus -- it's purely a matter of taste.  (In the world of plants, they have different taste from us vertebrate palaeontogists: most dinosaur genera have a single species, but many plant genera have hundreds, as in Quercus, the oak, which has 800 or so species.)

In fact, the only idea I've heard even vaguely floated for a "definition" of genus is precisely what you're talking about here: two animals are in the same genus if and only if they can produce offspring together.  This of course reflects Mayr's "biological species concept" (BSC), which defines a species as a population capable of interbreeding to produce fertile offspring.  ("Fertile" makes an important difference: because male ligers are infertile, you could never have a breeding population of them.)  I wanted to refer to this idea in a manuscript I submitted a while ago, but when I tried to find the source for this "definition" of genus, I couldn't find one, so it may well be that this notion has never been published.

So suppose we do define "genus" such that "two animals are in the same genus if and only if they can produce offspring together": does that solve your problem?  Not really: not only is it begging the question, but the "definition" doesn't work: imagine that it was determined that lions can reproduce with leopards, and leopard with cheetahs, but that lions cannot reproduce with cheetahs.  Then according to our definition, lions and leopards would be in the same genus, and leopards and cheetahs would be in the same genus, but lions and cheetahs would be in different genera!  In mathematical terms, the relation "can produce offspring with" is not transitive, so that this concept of "genus" is not an equivalence class.  Which makes it pretty useless as a taxonomic concept.

To throw a final spanner in the works, alert readers will have noticed that this flaw in the "biological concept of genus" also applied to the far more widely-used biological concept of species.  I don't really know what to do with that observation: I certainly don't have a solution -- I just admire the question.  I also don't know if this problem has ever been raised in print.

So: I've gone on for a long time, and answered slightly less than half of your question.  We didn't get anywhere near the destination, but I hope you enjoyed the ride.

Hi,

This is a real toughy.   Can I try to add a bit to Mike's answer?.  It is not directly answering your question but demonstrates one of the real dilemmas that affect biology to this day.  I suppose technically animals that are both members of the same genus but not of the same species should not be able to interbreed. If they can, then our concept of the species that they fall into has to change, so that they can become a single species, if the offspring themselves are fertile. 
Over time Rottweillers and Yorkshire Terriers might become 'distinct' dog species because of natural (height) barriers preventing breeding naturally, even though any offspring they did manage to conceive might be fertile. 

Different biologists classify species differently.  I will try and explain what I mean.

When I was at University, we had a lecture from a Botanist, a Microbiologist and a Zoologist.  Species are tricky things.  As we move from biological kingdom to biological kingdom, our concept of a species changes.  If we look at bacteria, viable, reproductive generations can result from gene transfer between bacterial organisms of the 'same species', that have genomes with radically different genes.  Two bacteria can transfer genes but have large percentage differences in their DNA, yet they are still considered to be the same species.  Is this sexual reproduction?  Probably not as we define it for more complex organisms but it is still mixing genetic material and causing an increase in diversity.

Plants also hybridise in a way that animals cannot, producing hybrids of parent species.  The hybrids are often fertile and produce normal 'hybrid' offspring. 

Animals seem much more restricted in what they can do, and the genes that they can mix to produce offspring.  Humans and chimps are 99% similar in their DNA.  For bacteria that would be more than enough to classify  two individual bacteria as the same species, but not for animals.  Humans and chimps cannot interbreed.  There have been suggestions that protochimps, and protohumans interbred for a long time after the divergence of the lineage leading to modern chimps and humans about 6 million years ago, but chimps and humans do not do so today.  No one has tried in a lab either, and nor should they, but the likelihood of a successful mating between one of the great apes and another is not likely to succeed.  A chimp is a chimp, and a human is a human.  There are almost certainly too many differences.  I can see no good reason for trying to see if it is possible either!
Likewise, mules (horse X donkey), Ligers (Lion X tiger) are infertile.  So the species that are being interbred are separate because the offspring is sterile.

So species exist.  The kind of organisms- plants; animals; bacteria- that we are looking at decides how we define the species, and how they can reproduce.

So, what is a genus?  That is a really good question, and I am not sure that I can give a good answer.  Genera are groups of closely related species.

Species are genuine taxonomic groups, kingdoms, and phyla are also now well supported as genuine natural groups, but everything in between (kingdom, phylum, class, order, family, genus, species) is fuzzier, so from class to genus is sometimes a little bit arbitrary. We have conventions and codes that we adhere to as biologists, but sometimes we have to erect sub-groups within these divisions to fit everything in.

I know this does not quite answer your question but it is a difficult concept to try and work out.  For animals, really only species should be able to reproduce, and that means producing offspring that can do so as well.

inter-genic reproduction (one genus with another) simply cannot happen by our definition of what an animal species is.

Well: inter-genic reproduction can't happen by one particular definition of what an animal species is -- but there are plenty of other definitions in town.  That's the wonder of species: there are so many definitions to choose from!