Sometimes when doing a background check you will see that a player played on a specific server previously. If you know the name of that server then you can check the server’s website for a ban list or forum should one exist.
An example of a ban list can be found here: https://alttd.com/bans/bans
Example of a ban list on a server website
Just type the player’s name into the ban list and hit enter. If the player is on the list it should come up as a result and you will be able to see their MC username and the reason they were banned.
Here’s another in a forum format. A ban appeal forum can also give you information on why someone may have been banned.
Example of ban information on a forum
Make sure to check the date they were banned, and compare that to the username on NameMC as it may have been a previous owner of the username that got the ban.