It's typically going to depend on your web host, and you should always contact them first before starting a large email campaign. If you're on a VPS (Virtual Private Server), because you share the same physical server space as other customers of your web host, there are typically no direct hourly rate-limits, but you have to ensure your VPS isn't overloading the physical server. A lot of email activity in a short period of time can cause a server's load to spike, and when you do that on a VPS platform, you can sometimes start causing issues for other VPS customers.
If you're on your own full blown dedicated physical server, then really your hardware would be your limit. If you're also running websites on this server, you don't want to send so many messages at once that your websites start to slowdown and lag due to the demand and load on the server.
phpList allows you to send rate-limited mailings to avoid problems with this type of stuff. To find out how many messages your server can comfortably handle, you'd want to watch the server's load with something like top
if it's a Linux server. Then keep an eye on that load over the time that you do your mailings, and adjust your phpList speed rate
to see what works best at keeping your load relatively low, while still pushing out a large volume of messages.
You also want to be mindful of where you're sending your lists to. For instance if you try to send to 5,000 AOL or Yahoo accounts, in under an hour, chances are they might start to greylist or even blacklist your mail server's IP address leading to extended delivery delays, or even total failures.