Mail.app with MailActOn (http://www.indev.ca/MailActOn.html) which I am not affiliated with and is not free. Use it for 7 email accounts, lots of RSS, all filtered carefully into one combined inbox and organized manually but quickly into local folders with that MailActOn app.
In my experience heavy use of Mail.app results in accounts that do not get checked for email, and show no connection error or indication of a problem. No email for an hour and I restart the app. I wish refreshing my accounts actually displayed the status of checking each account, it is very obvious sometimes that the refresh button does basically nothing or at least doesn't inform me in any way what it is doing...
Turn on the Activity Viewer: Window->Activity Viewer or ⌘-0. Shows each account being accessed and all related activity. Also allows actions to be cancelled.
In my experience heavy use of Mail.app results in accounts that do not get checked for email, and show no connection error or indication of a problem. No email for an hour and I restart the app. I wish refreshing my accounts actually displayed the status of checking each account, it is very obvious sometimes that the refresh button does basically nothing or at least doesn't inform me in any way what it is doing...