A series of great questions. First, the snotty fast answer:Did you ever encounter students like this? Did they go from being obviously dumb/low IQ to being smart? Or were they always smart and hi IQ - they just have a problem with one particular subject, until the light bulb goes off?
It doesn't take a lot of brilliance to get to a Ph.D. It takes mostly persistence and hard work, and an IQ in the 2+ Sigma range. Maybe for mathy fields, it takes closer to 3 Sigma, but still, that's not impressive. Almost all of us know some one who is substantially smarter than an average professor, but without the Ph.D. Probably that also indicates a lack of academic habits (and thought habits and prejudices), and a lack of a 10K academic articles read in one specialty. This is not entirely off the cuff. But it shouldn't be the whole answer.
The slower, fairer version:
This is every teacher's dream. This is why people go into teaching as a profession. Finding the kid who didn't understand it...who is having a hard time with a subject...and for whom your 7th explanation finally clicks, and the student begins to veritably glow with understanding...that is the God-moment for a teacher. If you've never felt it...you don't understand why people are teachers...and if you have felt it, you don't know how anyone can ever want to do something else. Mild exaggeration? Sure. The point stands.
This happens...sometimes. If a normal teacher teaches Algebra to 6 classes of 30 kids for a (4-month) semester, she may see this once. Maybe twice. It's generally worth all the crap for the whole semester for that once...if it happens. If it doesn't happen for 2 or 3 or 4 semesters, maybe she gets tired of teaching, and quits.
The Aretae explanation:
Interest is huge. Practice is huge. Cognitive readiness is huge. IQ is merely big. A smart enough, but slow-developing student who figures something out, gains an interest, and puts in effort to a Ph.D. 10 (or sometimes 20) years later is a real possibility. I've personally seen it done: Ph.D. in the real sciences by kids who were academically near dropping out of school but for an art teacher who believed in them, or a sports coach. And those weren't the 4+ sigma kids dropping out because school was hatefully slow and stupid...but rather kids who just weren't in the whole academic thing at 14.
Much more often, in subjects...I've seen folks who were more than a bit competent in a subject...but whose motivation and self-efficacy were destroyed by bad teachers. A particular teacher of pre-calculus many years ago did that to a number of people I knew. Her bad teaching broke the math spirit of an awful lot of otherwise math-comfortable students who never pursued the math that they would have been perfectly able to do, given a tolerable teacher and a decent book (they had neither).
With multiple thousand hours spent tutoring, I concluded that half my job was to straighten out the messes made by bad teachers. The other half was to do catch-up that no one else had noticed needed done. I mean, really...how DO you do fractions in a college level teaching-certificate pre-algebra course (or worse, when studying for the teacher-cert test) if your multiplication tables aren't solid? And how the hell do you teach the 6th graders when you personally don't actually understand the material at better than test-cram level? But I digress.
On top of all this, there's also the Pygmalion effect. Learning is massively (It's probably close to a 1 stdv difference) impacted by the teacher's expectations of a class. Futhermore, in school, IQs are not generally well measured, and it's not too hard to get a +1.5 Sigma kid placed in the slow group due to bored behavior problems. Once there, it's also often true that they're tracked in, and then you're really screwed.
As an aside...I'm something of an opinion outlier regarding subject-level intelligence. I think that the whole Howard Gardner multiple intelligences thing is 90% premium grade hokum. There is g, and there is interest and practice. I suppose there are (almost necessarily, given evolutionary understandings of the brain) various modules, and that verbal and spatial intelligence vary separately a little. But basically, g + practice (mostly determined by interest) seems to be far and away the best explanation for the picture we see.