I thought Everton was denied UCL qualification due to the team limit (actual limit is 5 I think), but they were allowed. My memory is clearly broken. I'm officially an old bag
To be fair, UEFA didn't really have a set rule back then.
Real Madrid won the CL in 1999/2000 but finished 5th, and the 4th spot that would have gone to Real Zaragoza was instead awarded to them. UEFA changed the rules so that CL winners had to earn a spot via league position to qualify so that teams weren't unfairly missing out.
Then in 2004/05 Liverpool did the same, but the rules stated that Everton had to qualify in their place. UEFA granted Liverpool special dispensation to defend the title, and the compromise was them entering the first qualifying round, and Everton took the proper spot in the final qualifying round. This saw Liverpool playing Total Network Solutions from Wales and FBK Kaunas from Lithuania on their way to the group stage. They were also not granted "protection" from being drawn against other English sides before the quarter finals, which saw them end up in Chelsea's group,
After that, I think they went back to the rule that gave the CL winners a spot in place of the lowest qualifier from their respective league, which is why Chelsea qualified in 11/12 and Spurs had to settle for Europa League.
When they introduced a spot in the group stage for the Europa League winner, the total number for a league was capped at 5, meaning if the CL and EL winners both finished outside of the qualifying spots, the team in 4th would miss out and be in the Europa League.
I believe this cap is going up to 6 with the expanded format from next season.