How do the rest of you handle email across a network?

I have a client with two Macs running Panther, both accessing an iiNet email account using Mail app. It is important to the client that both computers be able to read email, so they are each setup to delete email from the server after a week. For the most part the system works OK, except that on occasions there are frequent messages claiming that their login has failed, and it is necessary to re-enter the password, which invariably fails several times before succeeding again.

I am fairly sure that the problem arises from the two computers trying to access the same account at the same time, although it's hard to pin down a pattern. Do others have this problem on a similar setup? I'm trying to avoid the option of having to set up their own mail server (seems like overkill) but the problem is becoming very annoying, and it seemed to have but much less so in Jaguar. What is the best practice for this scenario?

I think we are doing something similar however the second machine is PC/XP using either webmail access or Mozilla Thunderbird.

Our ISP is Westnet, we leave the messages on the server indefinitely and manually clear them once a week or so.

We are regularly logged into the same account.

We have seen no issues arising from simultaneous logins that we can tell.

One thing however, originally Mail.app was set to check for new messages every 5 min, while this was the case we would often (once a day or so) see an error saying the mail server could not be contacted.
We set it to hourly and haven't seen a recurrence.

Cheers
Paul