Even in the cases of species that roughly meet the naive high school definition, it is at least sometimes the case that they can still interbreed, they just don't. Usually, this is because of geographic isolation. Take African and Indian elephants, for instance. They diverged long enough ago that they're morphologically distinct, not just genetically distinct, but they can still interbreed. They don't because they live on different continents, and they probably wouldn't if they were put together outside of captivity because they're intelligent, social animals with culture and learned histories who rely on not entirely biological cues regarding who to breed with, just as much as humans do.

In reality, we first categorized life into species because they either looked different or we found them exclusively in different places, and only centuries later did we attempt to figure out exactly why and how this was the case and reverse engineer some sensible definition onto the pre-existing categories, but it turns out there is no single definition that works universally and has zero exceptions. It's frustrating if you're a language pedant who likes clarity, but a lot of categories and definitions are like this.