AlyssaS - 5:18 pm on May 18, 2011 (gmt 0)
I don't have an answer for you, but I believe some of the moderators on this forum thread have said that, yes, you CAN hurt your site by pointing too many links at it. I tend to believe them.
I believe them too.
The site might be suffering from an anchor text penalty (where you have an exceptionally high % of external links with the same anchor text). The solution is to acquire new links with different anchor text to dilute the effect.
BTW, this is why it is worth making an effort to get natural links - they tend to come with a huge variation of anchor text, and provide stability to your site (it's hard for a malicious competitor to tank you by sending hundreds of spammy links with the same anchor text, as the natural links will dilute the effect).