Sorry for the delay here! I'm glad to hear all is well now. I've been doing some digging with one of our mail experts here, and this is what we've discovered:
The error you're seeing: "err: WSAGETSELECTEVENT = 0x20, WSAGETSELECTERROR = 10053", is something a number of Windows users of Outlook and Windows Live Mail have seen.
The 'nicer' error message at the end is just Outlook's interpretation of what occurred, not necessarily accurate. Interestingly, people have reported this error connecting a wide variety of mail servers. For example, some customers have it intermittently when using Gmail IMAP, often when switching between folders.
Regarding your question about the timing, we do mail backups in the hours around midnight PST, and the timeframe you gave was mid-day during regular business hours.
We're leaning toward this being a Microsoft issue with the mail client and how it does IMAP. Microsoft's mail program upgrades nearly all include upgrades that are not just to the interface, but also "under the hood" protocol stuff it uses to communicate with servers. Upgrading to a mail program less than 7 years old might be a good step. But after that, I would suggest asking Microsoft for assistance.
I wouldn't say this is definitely not on our side, but from the information we have based on our customers and a lot of random internet users, I would focus on the only thing everyone has in common -- Microsoft. If they find no problem, and/or have more details about what precisely is happening, we can definitely look into this further.
As for your boss possibly wanting to move mail service to another host -- yes, paying money for a service you value is much better than using one that comes freely bundled with web hosting. While the email service we provide free along with our hosting plans is usually plenty enough for the average shared hosting customer here (and more reliable than some other basic email services), it may not be the best choice for important/urgent business (and especially sales) messages and accounts. We take great pride in the services we offer, but admittedly, we know our email service is not the greatest, and is not yet quite as reliable as our hosting services.
For high-profile sites and accounts that are sending messages of dire importance, I would definitely recommend using a more robust third-party mail provider that can really promise certain levels of reliability. I understand how this may sound, and don't want you to lose confidence in our hosting services at all. This is simply my advice based on what you've shared, as I would hate for our mail service to cause similar issues for you in the future.