"only get a limited number of messages in my inbox"
Yes, that's exactly how the app works -- it syncs a limited number of recent messages, not all of them (what if it's a hundred K or more?)
The default is 25 - 50 most recent messages per folder.
To change:
Long press the account -> options and folders -> account options -> increase "Messages to sync".
The larger the value, the more work the app will have to do every time it checks mail.
So please be reasonable and try 100 or 250, don't jump to "thousands" right away.