I'm having the same problem, but it has been happening for a month or so. I use apple mail to manage my 10 different email accounts. Gmail, roadrunner, icloud, hotmail, and my website server emails. I get the triangle saying it's offline, emails show up again as unread after I trash them, they randomly go into archive. Sometimes when I send email it puts a copy in draft folder, but also says sent. Sometimes I see the triangle but when I look it says I'm connected.
This is happening with my husbands imac as well. He did not update to Maverick 10.9.1, I did. I use two different mac airs. It also happens on my phone, I use ios7.
I tried deleting all accounts and painfully added them all back. I also contacted my web host thinking it might be on thier side, but it's not.
I'm thankful at least I eventually get my mail. At least I think I am