When I was a kid the definition of a genus was a group of species that could interbreed. They've changed that for reasons I don't understand. Cats and ocelots cat interbreed and they do it so readily that the hybrids became a pet fad a couple of decades ago, yet they are now in different genera.
The reason that most species don't interbreed in the wild even if it's theoretically possible is that they have mating rituals. If they don't recognize each other's ritual they don't have the proper psychological response and they don't become aroused. Tigers have to claw each other to become aroused. That's the reason people had such a hard time getting captive tigers to breed, they were all de-clawed. I don't know whether lions do that, and if they don't they won't turn on the tigers.
I'm more familiar with birds. It's pretty easy to get the various species of Amazon parrots, macaws, grassland parakeets, etc., to crossbreed in domesticated environments.
The more gregarious and inquisitive an animal is, the more likely he will be interested in inter-species dating.