I like Cincy's point - upsetting #1 from out of nowhere is an impressive win, but there weren't especially high stakes going in.
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Very often, usually by teams that aren't used to being there, a team will overachieve and the moment they play a game with lots of hooplah, they fold and fold hard (see Illinois, 2001).
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Looking at their 3 most recent 'good' years (01, 89, 83)...
2001 - as soon as they enter the top 25, trounced by ranked Michigan
beat Purdue in a 20ish vs 20ish ranked matchup
trounced by LSU in the Sugar Bowl, despite being ranked higher
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1989 - actually beat #5 USC in the opener, but was only ranked 22, so while this counts, it was really just an upset
the very next week, top 10 matchup with Colorado, gets drilled 38-7
later in the year, back up in the top 10, vs #3 UM, loses by 2 TDs
won their bowl in a 11 vs 15 matchup vs UVA
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1983 - loses opener, but beats the shit out of #4 Iowa, 33-0 as unranked team (random big upset)
Ranked 19th, beats #6 OSU - another upset, but getting warmer
Here's the game we're probably looking for: #9 Illinois over #8 UM by 10.
Won the rest of their games before getting curb-stomped by a bad UCLA (as previously mentioned upthread)
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So I guess the "answer" in terms of the modern game is 9 vs 8 win in 1983. A relatively high-stakes game going in, with Illinois winning.
