I know it's a company because regularly they seem to mess something up and there is a post about it on /r/networking: https://www.reddit.com/r/networking/search?q=level+3&restric...

It's so common that someone even made a website about it: http://fuckinglevel3.com/

Remember when they nearly broke the internet back in IIRC '06, when they had that peering dispute spat with Cogent?

Telia de-peering with Cogent was bigger. And as others have pointed out, Cogent is frequently a bad partner in these disputes.

Given the number of peering spats that Cogent has been in, is it fair to blame the other party when one comes up?

It's not anymore, it's now CenturyLink, they closed their acquisition on Nov 1st.

Level 3 v Layer 3 ;)