Absolutely. But in practice, cognitive overload rears its head in many ways.
When most people think of an intuitive system, they think of a simple system as measured in the density of information on screen ("just show me what I need to know, nothing else") and the paucity of options you have to select from (Hick's law and so forth). But if you have to perform complex activities with such a system, the seeming simplicity results in actual complexity, since you have to do things like (for example) diving into a dozen nested layers of menus and screens to reach an option that was hidden there in order to (try to) reduce cognitive burden.
A different, equally valid measurement of simplicity (and cognitive burden) is how few transitions or state changes you have to suffer through to get to the state you want to be in. That is to say: how many steps you have to take to do what you want. And since there's a chance to make an error at each step, it's not even just a question of saving clicks, it can actually be more error prone.
I say these measurements are equally valid, but not that either of them are sufficiently robust ways of thinking about good design. I'm also not trying to downplay the impact of clear, intuitive design. The point I'm trying to make is that there is no formula for making designs clear and intuitive; it's actually quite complicated. In fact, in most cases, you can't even do it, you can only find a tradeoff between imperfect solutions.
Anecdotally, I have designed dense, complex systems for quantum computing experimentalists—people much smarter than me—and had them tell me "make it simpler!". I've also tried to make simple systems for biomedical researchers, and had them tell me, in so many words, to make the UI denser and more complex. There's no correct level of simplicity you can arrive at a priori, it's different for every case, and you have to reach the right balance through testing. If you don't do enough testing, your users become your testers. The consequences are not usually this grave.