http://bugs.winehq.org/show_bug.cgi?id=29876
--- Comment #11 from Adam Bolte abolte@systemsaviour.com --- I noticed that this isn't an issue for me anymore (installed Mass Effect 2 without issue), although my wineprefix is far from clean and I'm running Wine with a ton of patches. I suspect the job objects patch at #33723 is what fixed it.
Will add more information when I have time to do proper testing, but adding this here in case somebody else with more time wants to beat me to it.
Of course, it could also be game-related, or a fluke. It's also worth noting that ME2 didn't actually work (and appdb says it should - although the Origin build specifically appears untested).