Lions and tigers (and their crossbreed offspring) have fertlity levels well below what would be replacement levels in the wild. They are indeed speciated; they just aren't completely infertile. They almost never encounter each other in the wild anymore (although they probably used to before humans restricted their ranges). It's unlikely they would breed with each other absent confinement together by humans. Male tigers are particularly loath to breed with female lions, apparently considering them butt-ugly.
Lions also have some cross-fertility with leopards. This (and the fossil record) shows that the radiation of the big cats is a relatively recent thing.
There's a range of cross-fertilities visible in nature. It's what evolution predicts. Speciation is not an all-at-once thing. It takes time and cross-fertility in isolated groups drops smoothly rather than vanishing all at once.
We see every degree of relatedness in nature. Humans, before the age of exploration, were on the way to speciating based upon their geographic dispersal. It never got that far and now it never will because we've been remelding for centuries now.
How is it that you can say the same thing I say, but in half the words? I must babble.
Before anyone pounces to say that lions are in Africa and tigers are in Asia, there's a tiny forest preserve in India containing the last ragtag band of the Indian lion subspecies. I simply don't know if this area still has any tigers.
Lions used to roam southern Europe and much of the Near East. Tigers were once more widespread than now as well.