I am currently a student at OIST. The short answer to your question (right now) is “no.”
That said, we are doing pretty well and we are growing at an incredible rate. There was a recent online study by Nature (found here: A growing institute with big ambitions), that compared OIST to MIT, Harvard, Oxford, California Institute of Technology, and Tokyo. In this report, it was shown that OIST competes if and only if you normalize the data based on the institute size.
To be clear: this is incredibly impressive. OIST is less than a decade old, and to compete on any measure with some of the best institutions in the world is fantastic!
OIST is truly a one-of-a-kind institution that pushes the boundaries of modern research. Right now, it’s primary limiting factor is institutional size. We are growing quickly, but we are still missing key research in many areas (namely Computer Science and Mathematics). The department-less structure and independent nature of the research that goes on here has really opened the door to fantastic opportunities.
OIST is constantly growing and adapting. As long as it continues to do so, I have no doubt that we can match some of the best institutions in the world eventually; however, it is wrong to say that we are on par with Oxford, Harvard and MIT right now.