For example forum post notifications and personal message notifications. /message/output/email/message_output_email.php has some error handling/reporting in it. If emails are failing to send they should get queued or remain in an unsent state until email sending is fixed. For example if the email server is down, rejecting emails from Moodle or similar.

Email failure can be easily simulated by modifying email_to_user() in /lib/moodlelib.php