You are correct. That is the definition of ‘species’: the ability to interbreed.
Not exactly.
From Wikipedia:
In biology, a species (abbreviated sp., with the plural form species abbreviated spp.) is the basic unit of biological classification and a taxonomic rank. A species is often defined as the largest group of organisms in which two individuals can produce fertile offspring, typically by sexual reproduction. While this definition is often adequate, looked at more closely it is problematic. For example, with hybridisation, in a species complex of hundreds of similar microspecies, or in a ring species, the boundaries between closely related species become unclear. Other ways of defining species include similarity of DNA, morphology, or ecological niche.All species are given a two-part name, a "binomial". The first part of a binomial is the genus to which the species belongs. The second part is called the specific name or the specific epithet (in botanical nomenclature, also sometimes in zoological nomenclature). For example, Boa constrictor is one of four species of the Boa genus.
Tiger: Panthera tigris
Lion: Panthera leo
The fact that Tigers and Lions can interbreed has not made them into the same species.
Wikipedia article on Tiger/Lion hybrids: Liger
Horse: Equus ferus caballus
Donkey: Equus africanus asinus
Mule: Wikiepedia article on theMule
Now it is true that the definition above says "fertile offspring" and mules are typically sterile, but not always.
A few mare mules have produced offspring when mated with a purebred horse or donkey.[16][17] Herodotus gives an account of such an event as an ill omen of Xerxes' invasion of Greece in 480 BC: "There happened also a portent of another kind while he was still at Sardis,—a mule brought forth young and gave birth to a mule" (Herodotus The Histories 7:57), and a mule's giving birth was a frequently recorded portent in antiquity, although scientific writers also doubted whether the thing was really possible (see e.g. Aristotle, Historia animalium, 6.24; Varro, De re rustica, 2.1.28).As of October 2002, there had been only 60 documented cases of mules birthing foals since 1527.[17]
Brown Bear Ursus arctos
Polar BearUrsus maritimus
Wikipeida article on Grizzly and Polar Bear Hybrid