Maxflyer . . . . .
Can't see why not.
Even without going down the 'Thunderbird test route', just send a test e-mail from your Outlook account back to your Outlook account and copy to your AOL account, without connecting to dial-up (assuming broadband is connected).
Do I assume the question probably due to your Outlook (dial-up preferred) POP3/SMTP settings not being AOL related? Still shouldn't be a problem (I think; never know with AOL). I'm running Outlook, both on broadband and dial-up (same account), via an e-mail host completely separate from my ISP.
NB: Different but related issue arises sometimes when somebody using AOL's proprietary e-mail client suddenly wants to use Outlook/Outlook Express. Then it's a case of enquiring about and then enabling the AOL POP3/SMTP settings. But even if you have got these on your Outlook (suspect not?) it should work.