http://bugs.winehq.org/show_bug.cgi?id=12846
--- Comment #2 from Dan Jones djones9960@gmail.com 2008-04-28 19:55:14 --- (In reply to comment #1)
Does the demo have the same problem: http://www.ea.com/official/cc/generals/us/demos.jsp
No it doesn't have the problem. (the demo isn't available from the link you posted, BTW)
I've found it also doesn't happen if you download the cracked version of the .exe either. The copy protection would appear to be where the error lies (as I'm sure you suspected).