It's a war I'm particularly, acutely, aware of. Bureaucracy's purpose (in the modern workplace) is to keep idiots from blowing things up, trashing the company, or just generally doing the wrong thing. Some people are inevitable hired or promoted into jobs they cannot do well or even adequately; they are at best stumbling through a hit and miss relationship with success on every project they are assigned. Bureaucracy combats idiocy by inserting a known process or system that has been thought out beforehand into place where decisions are vulnerable to idiocy.
However, there are a lot decisions that are vulnerable to idiocy and erring on the safe side means creating a process for all of them. If that weren't bad enough, rigid process gets in the way of people who actually have a clue and could otherwise pull the company forward, forcing them to work at the same rate as their idiot counterparts. Nevertheless, this adjusts the range of employee talents towards a common, albeit lower, standard and allows a large company to operate with a large staff by treating its employees as a unit of resource rather than a fluctuating amount of failure and success. It drains the hope and life out of anyone stifled or stymied by it, but the upper management usually feels they have no alternative.
Even with warfare balanced between the two factions a Normandy of unyielding bureaucracy or a Russian winter march of phenomenal idiocy is still possible, but should not be feared. If either side were to actually win, stupidity or bureaucracy victorious, the company in question would self destruct. Speculative fiction paints a dark future where bureaucracy has gone berserk and people are treated as the lowest common denominator in every aspect of their life, but in reality such a corporation would be driven out of business by a competitor (unless it has no competition, for instance a government). Likewise colossal idiocy will also cause an internal meltdown of the company. No triumph of either side results in an entity that can compete in the marketplace.
But what if bureaucracy and idiocy were to meet on neutral ground and just talk, find out that they both have similar interests, they're both Pisces and like long walks on the beach? What it bureaucracy and idiocy decided to - oh nightmare of soul killing jobs - join forces? "But bureaucracy and idiocy are inherently joined", you might respond. "Bureaucracy is the method by which idiots work". While it's true that the bureaucratic method is idiot-centric, a method, or process, is usually prescribed by a higher level that has some confidence that this method will accomplish a job and protect the business. A unified front of stupidity and bureaucracy would prescribe nonsensical processes that would stifle every effort, brilliant and idiotic alike.
Would such a company fail much the same way as if they were simply blithering idiots? Not necessarily. Acts of sheer idiocy tend to show up in the Wall Street Journal and cause both investors and customers to scramble away. A company mired in misunderstood or inappropriate process cannot make colossal blunders with ease, since it cannot do anything with ease. Whatever the initial company course was, it will probably continue to be.
Would such a company be driven out of business by a competitor? Again, not necessarily. Small competitors might have a strategic advantage but lack the market share or sheer size to damage a company large enough to succumb to an idiocy-bureaucracy synthesis, while competitors large enough to be a threat would also be large enough to succumb to the same fate. An idiot bureaucracy run company might operate without any blatantly stupid fiascos for quite some time before it was outpaced by competitors.
Finally, is this all hypothetical or do such institutions exist? If this sort of thing were to exist it would be in a large corporation with a poor reputation that competes with other large corporations with poor reputations. Customers of such companies would be capricious about which service or product they pick, since it seems not to matter. They don't like the one they have, but don't like the competitor either. They may choose one company over another simply because the like the product color offered or the commercial catch phrase. Hmm, can I you think of any companies that fit this model?