Can't receive e-mails

[#1]

I use the Apple Mail program supplied with OS 10.4 (OS 10.4.9). Yesterday, when I connected and opened Mail, the login dialogue box didn’t appear, so I wasn’t able to access any new mail which may have arrived. I sent myself an e-mail to try to kick the system into working. Mail reported that the mail had been sent and I heard the 'whoosh', but I still haven't been able to access this message.
The settings in Apple Mail Preferences haven’t been changed. The Mail Connection Doctor dialogue box says ‘Could not connect to this POP server. Check your network connection and that you entered the correct information in the Account Preferences’.
The Ping option in Apple's Network Utility program reported;
mail.plus.net ping statistics
10 packets transmitted, 10 packets received, 0% packet loss
round-trip min/avg/max/stddev = 28.949/35.280/43.264/4.913 ms
So everything seems to be OK.
I’m stuck. My web access is intermittent. Some days I can't get online for more than a few minutes before the session is terminated, but today it's been reliable (no disconnections). The speed is still too slow (483.9 kbps for an 8 mb account, which isn’t what I’m paying for), but the web does at least work.
Does anyone have an idea what the problem is? Is there a preference in the System that I could throw away?
Thanks,
Ian

[#2]

We need to know what setting you have put in and then we may be able to spot the problem.
Can you get mail ok if you use webmail?
Your login name for mail would be blairyo
but if you have setup a mailbox you have to add this so if you have a mailbox fred you login with blairyo+fred

[#3]

I’ve not used Webmail before today, so I sent myself a test message. I’ve logged on twice, but the test message hasn't appeared in the Webmail Inbox.
I haven’t changed anything in Apple Mail. There’s just one e-mail address and one mailbox. In Apple Mail Preferences, the account type is listed as POP, the Incoming Mail Server is mail.plus.com and the outgoing Mail Server (SMTP) relay.plus.net.
Normally, to access my Apple Mail Inbox, I log on with my user name (blairyo) and password and the InBox appears. Unfortunately, the Log On dialogue box isn't appearing, so I can't access the InBox. How do I persuade the Log On dialogue box to appear?

[#4]

The name of the incoming server was mail.plus.net (I accidentally typed ‘.com’), but it still didn’t work.
It seems to be working now, but what a performance! I created a new account and after receiving a string of obscure error messages (‘relay.plus.net’ cannot be contacted on port 25’, ‘POP account appears to be offline due to a network problem and more), I succeeded in sending myself an e-mail. An earlier message, which seemed to have got lost along the way, then appeared out of sequence.
I have a horrible feeling that sooner or later, everything will go pear-shaped again. The Log-in dialogue box, which for two years has always appeared automatically, does not appear. Apple Mail has worked flawlessly for 2 1/2 years, so it’s strange that it’s suddenly started to misbehave. It would be good to understand why things aren't working properly.