Interbreeding is not a matter of "yes" or "no", there are varying degrees of "interbreed-ability" and they are a large part of what makes different "breeds", versus "sub-species", "species", "genera" or "families".
- Breeds of, for examples, humans (races) or dogs all interbreed easily, even eagerly.
Their offspring are sometimes the best of both and normally fertile. - At the level of sub-species interbreeding still happens in the wild, but perhaps with less frequency.
Here we might think of dogs & wolves or humans & Neanderthals. - At the level of separate species within the same genus, differences have now grown to the point where interbreeding in the wild is quite rare.
Here the example of Polar Bears & Brown Bears illustrates the point: classified in different genera previously, they were reclassified in the same genus (Ursus) as different species once it was confirmed they do occasionally interbreed in the wild. - At the level of different genera in the same family, the differences are so great that interbreeding approaches physically impossible.
Many examples can be cited: horses & donkeys, Asian & African elephants, lions & tigers, humans & chimps.
Yes, in some cases offspring can be produced by artificial insemination, but these are nearly always bizarre, infertile or otherwise unsustainable. - At the level of different families in the same order, there are no examples of interbreeding among more advanced creatures, that I know of.
But we should mention that amongst lower, simpler biological orders interbreeding may be less difficult than, for example, among primates.
You mentioned human/chimp chromosome count -- humans have 23, chimps 24 -- as a major factor.
Physically what happened was two smaller chromosomes fused in pre-humans, making a larger combined chromosome #2, restricting interbreeding -- even though overall human & chimp DNA is about 97% the same -- and making human linage separate from that point forward.
For more on the human-chimp genome project, see here.
