Hi Shane,
Kentico offers monitoring of bounced emails from out-of-the-box. It can be enabled in the Settings application -> On-line marketing -> Email marketing -> Bounced emails section -> Monitor bounced emails checkbox. You also need to set up a POP3 server, so it can work properly: https://docs.kentico.com/k10/on-line-marketing-features/configuring-and-customizing-your-on-line-marketing-features/configuring-email-marketing/configuring-bounced-email-monitoring
After that, once any email subscriber (contact) hits the bounced email limit, they will be blocked and won't receive emails anymore.