I have Office 365 for about 4 months now. I have my university email (Microsoft Exchange), my hotmail (now outlook.com) (Exchange ActiveSync) and my yahoo mail (IMAP) setup all on one profile. I have unchecked all of the email accounts in the send/receive group as well since I rely on push notifications.
I know that there's a bug with Outlook 2013 such that when both corporate/university email as well as an Exchange ActiveSync email are both active on the same Outlook profile, the EAS email gets disconnected from the server when Outlook is restarted. The only fix right now is to click work offline twice. Hope that MS would address this issue ASAP.
Here's my concern:
I have also setup my university email on my Android phone which supports MS Exchange. Whenever I receive mails to my university email, I always get the phone notification first then after 15 seconds - 1 minute (usually 30 seconds or less), Outlook 2013 will get the notification. For the past 3 months I have never bothered with this behavior since the push notification works. However,I was just wondering if anyone with Outlook 2013 or earlier versions with corporate/university emails both on phone and Outlook gets their notifications at the same time, or if Outlook gets the notification first before the smartphone does.
BTW, I haven't tested an only Exchange email account on another Outlook 2013 profile and see if that profile gets the notifications faster or as fast as the smartphone.