Thank you for coming to the Community with your questions! To clarify when you say the emails are not being delivered, you mean that even though they are assigned to an email list, the email does not make it to their inbox, right? When you see the reporting of the email does it say that the email "Sent" or "Bounced"? If its showing as bounced, take a look at this article on how to manage bounces. If you would like for us to take a deeper look, please email us at social_support(at)constantcontact(dot)com with a reference to this post, your username, and some of the email addresses you noticed are not receiving the emails. We would love to investigate this for you as we understand the importance of getting your emails delivered to all the correct recipients.
Hayley L Community & Social Media Support
If you find my post helpful, and it answers your question, please mark it as an "Accepted Solution" by clicking the Accept as Solution button in the bottom right hand corner of this post.