During Cup of the day it happened to me to exit and immediately re-enter the server (yes, I'm bad and hit the wrong menu), and after that I did a slower lap compared to my existing record, and that got recorded on the COTD leaderboard.
Needless to say that was incredibly frustrating. Even if switching servers, only faster times should be overwriting the time set for qualification.
Photographic proof at: https://www.reddit.com/r/TrackMania/comments/x2jhrn/today_i_learned_an_important_lesson_if_you_exit/