➕ I have the same exact problem as higginsd. Very similar setup with CentOs based server. To Recap, when i setup my imap accounts, they work fine; then, sometime within 24 hours, one of the accounts will stop working and I get "Cannot get mail The mail server is not responding". The account can only be fixed by deleting it and setting it up again.
What I have found is that it seems to be an issue with the outgoing server settings and how the iphone is storing that information. When i setup my imap accounts, I always set them up to use outgoing SSL port 465. Within 24 hours the port will change to port 25, which is for non SSL. Also, I have found that if you delete that account, when you set it up again, it will sometimes include past outgoing server settings from that particular account, causing me to have to delete the instances of the outgoing server for each account.
On my phone i have 3 email accounts, 2 accounts for 1 domain and 1 account for another domain. Since they are all on the same server, they all use the same SSL incoming and outgoing server name. Since iOS does not give you the option to name your outgoing servers, like apple mail does, this may be causing some of it.
I hope this information will help apple solve this soon.
Noting is worse than having to resetup an email account when you need an imortant email.