Apple supports the protocols for just about every combination of server I've come across. Some are easier to set up than others because they happen to use default security settings and ports that are well established. Apple is working on improving support for Microsoft's proprietary Exchange server according to the blurbs we get with the latest updates.
The fact that Copper.net says they are 100% compatible with Apple Mail tells me that you should probably check your Mail program for issues.
After taking a quick look at Copper.net's email server support page for Apple Mail I don't see any reason why those settings won't work with Apple Mail. They are old-school standard settings.
http://www.copper.net/kb/64-Email-Setup-Instructions---Mac-Mail/
The instructions are a little dated for Tiger, but still resembles an account setup in Leopard.
Of course, I can't actually set up an account with them, but I have no doubt it's a pretty straight-forward process. I've never had Thunderbird work, and Mail not work at the same ISP. Sometimes it's a little tricky for ports and SSL stuff, but it can be figured out. In this case it doesn't look like there's anything out of the ordinary with their server settings.
If Copper.net is so well regarded by somebody, then there must be other users here that have either had success or failure with their email system.