Where It All Goes Down
It shouldn't be surprising that a book with 300 characters has almost as many settings. But mostly, there is one aspect of all of them that brings them together. They aren't Russia.
The Émigré Community and Experience in America
Pnin's character can never be separated from the fact that he's a Russian émigré. So his relation to his adopted homeland is not the same as that of a Native American or even of an immigrant who was not forced to flee their country. For Pnin, America is pretty much a constant source of confusion.
The narrator says: "On the contrary, he was perhaps too wary, too persistently on the lookout for diabolical pitfalls, too painfully on the alert lest his erratic surroundings (unpredictable America) inveigle him into some bit of preposterous oversight. It was the world that was absent-minded and it was Pnin whose business it was to set it straight. His life was a constant war with insensate objects that fell apart, or attacked him, or refused to function, or viciously got themselves lost as soon as they entered the sphere of his existence" (1.8).
So for Pnin, it's as if this strange new land is always attacking him. Stuff just keeps going wrong, and he just can't seem to get the handle of living in the United States.
Waindell (Vandal) College (Wellesley College/ Columbia University)
Nowhere is this not-being-able-to-get-a-handle thing clearer than at Waindell College. This fictional university was apparently modeled on some kind of combination of Wellesley College and Columbia University. Nabokov himself taught at both of these universities, and based on their portrayal in Pnin,we would assume that the experience was not entirely pleasant. No wonder he has Pnin mispronounce the name as Vandal College.
Waindell is an all-American institution, which is exactly what makes it a terrible place for our poor Pnin. The narrator describes it: "He taught Russian at Waindell College, a somewhat provincial institution characterized by an artificial lake in the middle of a landscaped campus, by ivied galleries connecting the various halls, by murals displaying recognizable members of the faculty in the act of passing on the torch of knowledge from Aristotle, Shakespeare, and Pasteur to a lot of monstrously built farm boys and farm girls, and by a huge, active, buoyantly thriving German Department which its Head, Dr. Hagen, smugly called (pronouncing every syllable very distinctly) 'a university within a university'" (1.4).
Can't you just taste the disdain? So Waindell's not a top-ranked university, it's kind of fake, and its idea of itself as a splendid educational institution is incredibly bloated. Just the way the narrator talks about attempting to teach "monstrously built farm boys and farm girls" such things as Aristotle, Shakespeare, and Pasteur sounds completely ridiculous.
It is out of exactly this kind of overblown ego that the majority of professors at Waindell get their hatred for Pnin. They see him as some kind of intrusion into this all-American institution, and feel that he doesn't belong. Other professors make fun of him, avoid him, and call him an idiot. Not because he's actually done anything wrong, but because he's kind of strange and foreign.
Cook's Castle
Then, on the other hand, there is Cook's Castle. This is the place where Pnin visits some friends during the summer, and it's the first and pretty much only time in the entire novel that Pnin seems to be right at home. And it makes total sense, since he is surrounded by other Russian émigrés.
The narrator describes it: "Within, the diversity was as great as without. [...] In the half a dozen rooms of which each of the upper floors consisted, and in the two wings in the rear, one could discover, among disparate pieces of furniture, some charming satinwood bureau, some romantic rosewood sofa, but also all kinds of bulky and miserable articles, broken chairs, dusty marble-topped tables, morose etageres with bits of dark-looking glass in the back as mournful as the eyes of old apes. The chamber Pnin got was a pleasant southeast one on the upper floor: it had remnants of gilt paper on the walls, an army cot, a plain washstand, and all kinds of shelves, brackets, and scrollwork moldings" (5.4.1).
In other words, Cook's Castle is a kind of giant mishmash of a building. A bunch of elements from all different eras were taken and smashed together to create one giant mansion.
If you think about it, what could be a better place for a bunch of Russian émigrés than a building that doesn't actually adhere to any place or time? After all, they aren't really at home anywhere. They aren't at home back in their homeland of Russia, because of the Bolshevik Revolution. They also aren't at home in their new lives in America. They also seem to be living in some kind of mishmash of the past and the present. So why not a building that withstands countries, eras, and architectural styles?
Thinking about it this way, it's no surprise that Cook's Castle is the only place that we see Pnin actually enjoy himself and win at something. It makes sense. After all, it is a setting made just for him.