I assume you don't have a brand?
Perhaps having a 'brand' is good enough to rank (for a variety of brand-related factors, eg trust which translates algorithmically somehow)
I can speculate for days but it is evident that they are doing something right, whatever it is, and that you need to improve something.
You acknowledge your quality isnt good, so you should start there, (if indeed you it was panda that you were affected by). My understanding is that you will need to do something to change your content from bad to good in order to rank. You got hit by Panda and they didn't so the burden is on you to do something.
This type of things happens all the time. Trust me, the same thing happens in my space.