Do you mean it doesn't get some of the recent messages?
Sounds like the server shuffles their order, so they're not ordered by date, and the app doesn't get all recent messages when loading only a portion of the server side message list.
Thunderbird (and other desktop mail apps) have the luxury of downloading everything, and will get new messages even if the server puts them somewhere in the middle of the list.
Please long press the account, choose Options and Folders, then Account options, and increase the "messages per page" to a larger number (250 to start).
Oh, and how messages do you actually have in this account, on the server? If you have web mail access, that would an easy way to check.