As a professional sysadmin for a (not just web) hosting provider, any time I've run into Fedora on a server it has been an indication that:

1. The client was running something obsolete and unmaintained that would not survive an update. This would generally be a version of Fedora 2-12 versions behind current
2. Overtime was in my future as rolling updates broke their business a critical application
3. The system was set up by a client's family, friend, or other nonprofessional sysadmin who would (or could) no longer support the rickety framework they had built on top of it, or
4. Some combination of the above

I could imagine it working in a devops environment at a company with a real development team that also happens to understand what sysadmins are for, but haven't run into that in practice.

Seriously though, for a server you need something where security updates don't end the day a newer version is released. LTS releases and security backports matter for stability, and you don't get that with Fedora.

Edit: To be clear, I saw all of those things on other distros as well. I just can't remember a single Fedora instance where I didn't see one or more of them.