I am having the exact same problem. It has been persistent for the last two weeks.
Brief summary:
I recently purchased Mass Effect 3 (having not played it yet) and then decided to replay the entire series. I currently have Mass Effect 1 and Mass Effect 3 fully downloaded including DLC, but am unable to download Mass Effect 2.
Every time I attempt to install Mass Effect 2, it fails to start the download, and then crashes Origin.
Here is what happens in each step:
1. I hit [Install]
2. There is a brief loading bar labeled, "Installing..."
and then either
A1. Loading bar disappears
A2. [Install] button reappears
A3. Mass Effect 2 does not begin downloading
or
B1. Origin Crashes and requires a restart
This problem began after an attempted download of Mass Effect 2, which initialized successfully but caused Origin to crash at some point during the download. Since then, I have not been able to successfully initialize a download.
This has been persistent for the past two weeks. I have submitted a few bug reports and numerous crash reports, but there has been no changes nor response.
Additionally, in case it matters, I purchased Mass Effect 2 sometime in 2010. Added to the library on October 15.
I have already uninstalled Origin and reinstalled the software. Continues to produce failed results.
Hoping these forums might turn up a suggestion or, even better, measurable results. Though I'm beginning to lose faith in EA's customer service...
P.S. I've already finished my Mass Effect 1 play through I'm in desperate need! Please save me from my torment!!