Honestly, it depends on the computer, and the actions taken beforehand. If Mike tried to login to his account right before the password is changed, then the page he tries to login on will have cache files stored through the client-side that will make it seem as if he is still entering in the wrong password. This is why it’s effective to refresh the page before trying again.
Alternatively, the system in which the information about the site is stored may be trying to refresh the information. It is not uncommon for larger sites to be systematically separated across several servers to either a) prevent a massive shutdown in the event that one system is compromised or b) make it more efficient for users that are logging in at a proximity closer to a particular server than the “mother server” (faster ping to closer operating system). Updating information from the help desk would take anywhere from a few moments to a few minutes to sync across the entire band depending on the efficiency of the system, so this could be the issue as well.