what do they mean by "messages were being sent "through the webspace""?
It doesnt seem logical that how the messages are sent would cause the problem, if that where the case, the problem should happen on the first email. The first email is sent exactly the same way as the 189th email.
The only real difference between the 2 is the number of emails between them.
I know 1 and 1 has a 99 message at a time limit on their SMTP server, so you may need to set the email settings to limit to 99 message every hour.