How to destroy people's interest in updating documentation
Here is one of the less obvious perils of outdated documentation:
Suppose that you have some documentation that is out of date, but not in an obvious way; for example, you have an out of date network layout diagram. Since it's not obvious you don't realize this right away, so you keep on updating the network layout diagram when you make changes to your actual network.
Except that faithfully updating an inaccurate network layout diagram is relatively pointless. When you realize that it is incorrect, you are going to have to re-check most of it anyways, or at least spend a bunch of effort to reconstruct what sections are trustworthy.
This peril of outdated documentation is that updating bad documentation is wasted effort. (Fixing bad documentation is not, but that's a different thing.)
Since updating documentation takes time that you could be using for other things, and it's generally not fun, it does not take too much time to be wasted this way before people stop doing updating documentation entirely. Why do annoying wasted effort, when you could be doing something that's actually productive and useful? (Especially if you did the work thinking that it wasn't wasted effort, only to find out later that what you thought was productive work, well, wasn't. People really don't like that.)
At first, this effect will probably be limited to documentation that is highly suspect. But I don't think it takes much bad documentation before people more or less give up totally, because it is too heartbreaking to waste time this way and they can't stand the idea of it any more; you will lose the culture of documentation. At that point, you can stop talking about updating documentation and start talking about reconstructing it from scratch.
(This is where local wikis are perhaps less than ideal, because at this stage what you really need to do is pave everything so that there is a clear line between 'done recently, can be trusted' and 'is old, do not trust until it has been redone'.)