Snow leopard " MAIL" Not working

All my settings are correct.
And when I open connection doctor, at first,
all incoming and outgoing accounts are green.
Then after a moment or two, they all go red.
There was something in trouble shooting that said,
to delete all accounts and start over?
I'm at a loss as to what to do.
If I do that, will it delete my inbox @ aol?

I forgot to mention, that my internet is directly connected to the cable modem.
And the internet connection is fine.
Perhaps it is a setting from time warner?
And my accounts are set up as IMAP.
Thank you for responding, it would be nice to use mail instead of
logging in to each account @ aol.
I will back up the mail folder as you suggested.
And I do have time machine.
Thanks, John

A drop down window opens with the options of: Quota limits, mailbox behaviors,subscription list,
and summary.
under subscription list: this account is offline.
Summary: incoming: imap.aim.com-ssl:on / outgoing: smtp.aim.com-ssl:on
I don't understand why this is so difficult to fix.
They should have a option to reset or reactivate.
Sorry about the lame comment.
signed, frustrated, John

Sorry it took me so long to respond.
The port in use is 993.
What port should I be using?

John

When I set up mail the system turned on ssl.
I tried all three accounts with ssl off and that did not help or work.
I have forgotten how I found the port setting.
Can you tell me the path to edit the in and out mail ports.
Thanks a bunch,
John
P.S. Troubleshooting suggested to do a port scan for the mail host.(AOL)
So I did and they came up open tcp port:25
As of today the three accounts are still offline, but connection to internet is green.

I was able to change the incoming and outgoing mail ports.
And turned ssl off.
Connection doctor shows all accounts green.
Activity monitor shows alot of different types of communication to the server.
But the main inbox for one account has not updated emails.
The sprocket spins but no new messages.
There are 114 recovered messages under the recovered tab.
And thinking perhaps 3rd party software, like SOPHOS or iantivirus,
may be a cause.
I uninstalled them?
I'am so at my wits end with this.
Appreciate your tips and suggestions,
Thanks John

Here is today's latest:
All accounts are green, activity monitor shows
all kinds of communicating with server.
I have tried all of yours and others suggestions on which ports to use,
both incoming and out going.
With ssl on and off.
Here are my current settings:
imap port 143, no ssl
smtp port 25 no ssl / ALL MAILBOXES ARE GREEN /
But the one account, that lets say I receive 10 emails a day is not updating.
For example, when I log in to this email (From Browser) let's say I have a 100 emails in my inbox.
When I look at OSX mail there are only 25 emails.
And the last new one was from march 2010.
This account in particular has the spinning wheel, but the one thing that does update
is the recovered messages folder.
It has repeated one email, the same email over 100 times.
What's up with that.
My only hope is to delete all accounts and start over?
Or can apple email set up reset, if I try that?
Or never use it again.
Like I have said there is no rhyme or reason on how to fix this.
I thought once all accounts went green I would be good to go.
Not so.
End of wits,
John
P.S. Thank you to all for your help.
Any final tips, let me know.

Answering your question,
that option shows take account(s) offline.
I'am closing this thread, due to I'm done.
For over a week I have tried everything
to no resolve.
Thank you,
John