I presume its appropriate to continue this post even though we are well into El Capitan? Please advise if I should be writing under a new post!!
The version of this issue that I am experiencing is different, in that it relates to the hardware product I am using (MacBook Pro Retina), rather than the OSX version.
I purchased a MacBook Pro Retina in Sept 2015 to supersede my two ailing/slow MacBook Pro's (late 2011). Since that time, I have had the opportunity to observe the behaviour of three MacBook Pro's on the same wi-fi network, with exactly the same account settings (note with Apple's support we tried every version of setttings possible), the same three email accounts (1x gmail, 2x third party hosted emails), and exactly the same software updates (initially Yosemite, and now El Capitan).
On the new Retina (while still on Yosemite on all machines) I started getting the IMAP servers intermittently going offline, with incoming server and outgoing server alerts (not always, but several times per day). When those alerts were showing on the Retina, it would take several minutes for emails to send or receive, and emails would sit in the outbox as if they had not been sent. But they would all eventually send/receive. At the same time, the other two MacBook Pro's would show no issues (alerts) whatsoever, and be able to send/receive with no problems at all. Curiously, if I sent an email from the Retina to myself, it may show as being in the Outbox of the Retina, but still be received "instantly" by both other MacBook Pro's. I also had another clearly related connection issue, in that iTunes would regularly bounce up and down with a message “We could not complete your iTunes store request. The network connection timed out” - this is regardless of the fact that I had not purchased anything, and iTunes was shut down.
Apple suggested I wait for the El Capitan update, but exactly the same behaviour. In the end, after lots of Apple Care advice, Apple agreed the issue was restricted to the Retina machine and decided the issue must be hardware related, and so replaced the MacBook Pro Retina in December. For the first three weeks my newest Retina has been perfect (problem solved!!!), but two days ago - the "identical" email/connection issues. Apple tell me "today" that "an issue has come to light with the software that had caused similar issues with multiple users. This is currently under investigation by engineering...keep an eye out for incremental software updates..."
As above, this is the same issue others are experiencing on this thread, but the common link for me is the MacBook Pro Retina. Please advise if that requires a separate thread. In other respects, any thought would be greatly appreciated. Regards