After taking a look at WineHQ, the problem lies with Wine's network libraries. Recent releases of Wine, that are not yet incorporated into Crossover, resolve this issue.
I'm using the Steam version of Company of Heroes and I play it in offline mode.
1) Disable your internet connection.
2) Start up Steam in Offline Mode.
3) Start up Company of Heroes.
4) Company of Heroes will ask you where you purchased your copy from; select "Steam".
5) Company of Heroes will ass you to insert the CD/DVD; press "Ok". Steam will allow you to play even without a CD/DVD.
6) Enjoy!