Are you seeing this error in your logs? Chances are your client has triggered a fail safe in Exim that restricts them from sending any email out of your server. This event is triggered when an account sends email to addresses that fail or defer more than the set limit. So for example, an account sends out 10 emails in one hour. 5 of these messages are to addresses that fail or defer and the other 5 deliver ok. That means that this accounts fail/defer percentage per hour is 50%. If your limit set within the WHM is 50% or lower mail will now be rejected when that account sends more email.
Here's how you can change the threshold or disable this feature!
Maximum percentage of failed or deferred messages a domain may send per hour