If you are using our shared email servers then yes we do. We will not tell you what the hourly limit it but it has never been exceeded by normal day-to-day business emails even from extremely busy web sites and email users.
The only time our limits have been exceeded is when someone attempts to send a mass email which is not allowed on our shared email servers.
We also run an anti-hijack system to try to protect our servers from being used by spammers. This system "kicks-in" after certain threshold have been exceeded. When it is first triggered it temporarily holds-back the outgoing emails for a certain period. If other thresholds are reached in terms of numbers of emails, all emails for that account will be stopped from being sent out.
As a user you will not know if our anti-hijack system or email thresholds have been exceeded, and our email servers will continue to accept incoming emails for delivery to other accounts. However, our email server will inform us when our thresholds have been exceeded so we are always aware of what is happening on our email servers.
We do this to keep our email server "clean" and to try to ensure our email servers never end up on an anti-spam blacklist. This means that emails being delivered from our email servers are accepted as coming from a spam-free system and are much more likey to be accepted by the recipient without any problems.