My understanding of the reason behind this plot is that, empirically, it's rare for code to have more than four independent instructions that can automatically be recognized to be independent by the processor. Therefore, even if the processor is as smart as a human and able to exploit any available ILP, the available ILP itself is limited to around 4 independent instructions. Is this correct?

As a followup to donquixote's comment, I think the explanation that the professor gave in the lecture is similar. It is basically due to the fact that most program is just a sequence of instructions, so it doesn't help if there is nothing to parallelize even if we increase the ILP