BBCnews is carrying this article summarizing new discoveries about the evolution of cats. By studying mitocondrial DNA in living cats, scientists have determined they evolved from a common ancestor in present-day Asia some 11 million years ago:

The Panthera lineage, which includes the lion, jaguar, and tiger, emerged first. This was followed rapidly by a group of three Asian species - the bay cat, Asian golden cat and marbled cat; three African Species (caracal, African golden cat and serval) and the path that led to the New World ocelot.

If I recall my human origins classes correctly, this would put cats evolving at roughly the same time as apes, and in more or less the same areas. No wonder the wee beasties always look like they want to kill us!