What if it's the other way around: what if the feeling of everything being interconnected is an illusion?

When Cartesius said his famous "Cogito ergo sum" that's pretty much what it meant - you can be sure that your mind exists, but not much else beyond that.

And, if you think about it, why would everything in the world be interconnected in some meaningfull way? For what we know universe is just a big bag of energy and particles interacting with each other in the random way. "It's all connected" is a cry from the human mind desperately trying to make sense out of chaos.