Hi Kostya - I am also experiencing the same problem on my S3. I have a pro key, and 4 POP3 accounts set up (2 gmail, 2 work). I've been noticing this strange behaviour for a few weeks now, but having trouble finding the cause. I can see that I'm not the only one! :-)
When I'm on wifi, at home or elsewhere, there seems to be a weird message delivery problem, where only some of the messages are coming in on my various accounts. At times, it will take several scheduled syncs (pulls) before it will get the message, usually several hours late if at all. Some just don't come in. When I eventually get back to my pc, I find that my client (Outlook) has a bunch of messages that AquaMail never managed to pick up. Outlook is set to leave everything on server for all accounts - and this snafu happens even if my desktop is powered off (eg - AquaMail is the only active client accessing my mailboxes).
I just experimented with this - I had the S3 on wifi at home, there were around 12 messages that Outlook received this morning, but AquaMail didn't get them, even tried several manual syncs. I turned off wifi, and poof, all the messages instantly came in via LTE during a manual sync.
Other times, if I open/read a message in AquaMail, and leave it in the inbox, when I get home it never makes it to my Outlook clients (but they are there in my gmail / webmail). I realize there are differences between POP3 and IMAP, but I think AquaMail is flagging some messages in a certain way, where Outlook never sees nor fetches them. This has never been a problem for me before - I can have Outlook running on pc and laptop, and I was with BB before the S3, but with everything set to 'leave on server', I'd never miss a beat - open something on the pc, and the other devices would still be able to fetch it at the same time or later.
I really like AquaMail, and hope you can shed some light on fixing this problem.