Because of [...] the massive number of tests students take I bet that the smartest kid in every high school knows that he/she is at least one of the smartest 3 kids in his/her school.

That's probably close to true in the US, at least by college admissions season -- there are only a few merit-based scholarship packages that are open to anyone going to any university, and if you've landed one of them, or even gotten close, you can be pretty confident that you're if not the smartest kid in your school then at least in the 98th percentile or so. (There is some noise.)

I think it becomes a lot less true at percentiles below the 95th or thereabouts, though. You'll have gotten standardized test results, yes, but if I'm remembering my own high school years right, they'll likely have been perceived (not entirely without justification) as utter bullshit. Grades are better correlated with conscientiousness than IQ, and you'll probably have gravitated towards students close to your own intellectual caliber, so social proof won't be helping you much. All told, I think I'd expect high school students that aren't obviously more than two or three sigmas out to perceive themselves as much closer to average than they are.