Correct on Chilton. Looks like Sternvogel put up a Fillmore Junior High - same thing?
Nope. Different school, different show.
Bayside High - Saved by the Bell
PennBrook University - Boy Meets World
For Babysitters Club/Stoneybrook Academy–did any of the characters go there? I know their main school was Stoneybrook Middle School. But then again, I think characters like Shannon or Bart might have gone there. (Ugh, I’m a dork!)
Correct. (Someone else got Saved by the Bell.)
Horace Green Preparatory School: School of Rock
Sorry about that.
Some new ones:
John Adams High
Mars University
McKinley High School
ETA: Worthington is Dawson’s Creek.
Dudley Garrett gets another point for Langley!
Freudian Slit, you get a point for Dawson’s Creek. 
I don’t know about Stoneybrook Academy: I presume in that fictional world, all the institutions are named for the town. It’s a pretty common convention.
fachverwirrt gets another point for School of Rock.
Academy of Computer Sciences — 1980s t-t-t-television
Furinkan High School — anime
Huber Elementary — Cartoon Network
Io House — British television
North High — contemporary television
Oats Military Academy — duuude
Seaview Middle School — if you had a preteen daughter you’d know this
St. Bartleby’s School for Young Gentlemen — a foul clue
Thomas Aquinas Private Girls’ School — anime
Westbridge High — talking cat
Whitney High School — 1980s television
Winslow High School — contemporary television
Wordsmith University — Russian novel
Also Boy Meets World.
I’m going to take the “duuude” clue for Oats Military Academy for…the scary military school that Ted “Theodore” Logan was almost sent to in Bill and Ted’s Excellent Adventure.
ETA: Yes for John Adams High.
Arnold J. Rimmer was from Io in Red Dwarf, so that’s my guess - but I don’t think it’s right, I’ve read most of the Red Dwarf books and it’s never mentioned. Besides, it wouldn’t really make sense to name the school after the whole moon, if the school was on that moon, would it?
Yes, Freudian Slit, that’s the one. Duuude.
And yes, Dead Cat, Red Dwarf: there was a mention in the TV series. As usual, Rimmer was complaining how his brothers got to go to the really good military space academies, “while [he] was lumbered with Io House.” Not sure if Io House is the name of the school or a fraternity in the school, but it’s listed as a school.
Die Hard 3?
Huh, cool. Red Dwarf never made much sense anyway :). Great show, though - the books are even better IMO.
As jayjay guessed, Fillmore Junior High was The Brady Bunch (Marcia practiced her “F-F-F-I-L, L-L-L-M-O, O-O-O-R-E – Fillmore Junior High!” cheer in one episode), while Tom Landry Middle School is the educational institution Bobby Hill attends.
DudleyGarrett correctly tied Walt Whitman High School to Room 222 – the show’s title referred to the classroom staffed by Lloyd Haynes as American History teacher Pete Dixon.
Hint: Harding High and Grant College are from the same show (although a different show than the ones previously mentioned).
An obvious shout-out to William Daniels, who played the principal. Daniels had previously played John Adams in 1776, both stage and screen versions. (Thank Og it was John Adams High rather than Knight Industries High…)
Some more:
Carver High
Disco Tech (high school)
Roosevelt Franklin Elementary
Devon Academy
Wright-Dobie School for Girls
Sesame Street
One more:
Paul Westerberg High School
One more I’d meant to include: North Manual High. So…
North Manual High
Carver High
Disco Tech (high school)
Devon Academy
Wright-Dobie School for Girls
Yes, Roosevelt Franklin was on a gritty stretch of Sesame Street.
A couple more:
Rydell High (an easy one)
West End Grammar School (Even googling didn’t turn up a good hint here- kudos if you get it!)
Hirsch College