One reason you might not be able to download email is that you may be accessing the mail server too often.
If the mail server is accessed from one IP address more than 15 times in one minute the mail server considers it an attack and will shut down the mail server. This is a function of the mail server software to protect the mail server from attacks and cannot be changed. The IP address is then blocked so that our other customers on that server can access their email.
What you will need to do is find a way on the client computers to ensure that the email server is accessed 15 or less times in a minute from the same IP address. We recommend batching the email download requests in groups of 10-15 at a time, with at least a couple minutes between each batch request.