When I was at music school, we had a coven of enthusiastic drummers, and almost every week one of our members would see some note-worthy drummer at a club and dutifully report the next day that the player had '4-way independence.'
But there's a problem with that assessment. In order to have all four limbs move truly independently, each one would need to have its own brain, and extra brains are pretty rare. Octopi have a separate 'brain' for each tentacle, and each can go about its business more-or-less unsupervised. But we poor humans have to make do with just the one brain to manage four disparate activities.
Despite evolution's short-sightedness, we've managed to get by fairly well in mono-brain mode. Take the old saw about walking and chewing gum at the same time. While these are completely independent tasks, they are simple repetitive movements that require almost no brain power. Once we start walking and chewing, we switch to auto-pilot and walking just happens! If we want to stop or change direction, the brain steps in and manages the change. Then it’s right back to auto-pilot.
We drummers take the walking-with-gum thing a lot further. We can keep a syncopated bass drum pattern going while playing a different pattern on a cymbal. We can then add some contrasting snare drum work. Throw in a hi-hat and it seems like 4-way independence. Four-way? Yes. Independent? Well, no ... not really! As with walking, one, two or even all of those actions are on auto-pilot most of the time. We can make little changes here and there, but then auto-pilot takes over once again, thanks to your remarkable nervous system.
Independence mainly relies on ostinatos and (so-called) multi-tasking. An ostinato is a set pattern that is repeated by one voice. Your ride rhythm, for example, is an ostinato, and it is played pretty much automatically, freeing the other limbs to do other things.
As for multi-tasking, humans can't really do this. What we actually do is quickly switch the focus of our attention, and our ability to do this is limited. We work on one thing, set it aside to work on something else, then put that task on hold while turning to a third or revisiting the first. For many tasks, we can switch in a microsecond. So I can leave that cool ride rhythm unattended while I do a bit of thing on the snare, and then my attention can go back to the cymbal.
Of course, all this wonderful co-operative activity is taking place behind the scenes in the brain -- what we often call muscle memory -- and for a lot of tasks it’s very useful. But for playing music, it's absolutely vital.