I won't name names, but a company that I worked for was very web-like, which felt good at first because it seems more "fair" or democratic, but it lead to constant confusion, infighting, communication failures, etc., and I discovered that it lead to two forms of employees: hogs that managed to eat more from the trough through clever maneuvering, and "homer simpsons" who figured out they could be lazy and not suffer serious consequences since their title was nebulous and the chain of command was diffuse(an individual with poor performance didn't reflect that badly on a superior).

The team that I worked on functioned mostly as a web, and that actually worked well because we were only 6 people at most, but an entire company functioning as a web is a nightmare.