POP or IMAP? If IMAP, as sgunhouse said, check the settings on the "Incoming" tab in the account's properties. As long as you don't have "low bandwidth mode" set or "do not download attachments" set, you shouldn't have to choose "fetch complete message". That should be the case whether you have "make messages available offline when I click them", "make all messages available offline", or "do not make messages available offline". But, things in Opera Mail (like search) will work better with "Make all messages available offline", so set that anyway if you haven't.
If IMAP and none of that works, you can try deleting the IMAP account and adding it back it to see if that fixes anything. Or, you can close down Opera Mail, rename the "mail" folder (see "Menu -> Help -> about Opera Mail" for the location" and start fresh.
If none of that helps, we'd need to know the mail provider you're using.
