There are three reasons this might happen:
Groups
Do you have permissions to see the group that the trainee is in? If you are a Manager and can only see the Dunedin group, but the trainee was only invited to the Napier group, you won't be able to edit that trainee. You might be able to see reports that have them in, but you won't be able to search for them, validate their course, etc.
Expired course licence
If the trainee's course licence has expired, you can find them in the expired/deleted drivers list. You might not be able to search them, though. You can use ctrl-F (or Apple-F) and search for their name in the browser.
Account permissions
The main account holder must give you permissions to edit trainees, otherwise you won't see them.