If I know somebodys IP address, can I find out their email address?

NOPE.
It's not that easy.
Yes it may be possible from an IP to find the mail server that has been used, but to then find a single user on that server isn't going to be possible.
You could guess once you know the server.
For example if the IP shows as Homail then you could guess the address from that.

If the IP you have is just from their net connection then you don't have a hope.

The simple answer is it's impossible. Under the internet privacy acy under Clinton's administration the furthest you can track back an ip is to the internet service provider, beyond that point you have to have the authority or permission to request the full trace. Also, if the email account the person uses is online i.e.. Hotmail or Gmail or Yahoo, then the ip address of the computer they were using may not even correspond to a specific email account or an account at all.