What's with the air-quotes? Is it a rule or not?
On the surface, the 3-click rule is just plain wrong: the amount of clicking alone has very little to do with the users experience. What does make a difference is the so-called success of those clicks.
There's no reason that some content can't be buried on your site as long as there is a clear and followable information scent for your visitors. Let's say I visit your site and it takes me 10 clicks to find your information. If each of these clicks was on a clearly labeled button or link that seemed to be leading me in the right direction, I'm probably not going to mind it all that much.
Now if your site is poorly designed and organized and your links/menus are unclear, half of those 10 clicks could be on the back button. The content itself could be accessible with one click, but if I'm unable to find the trail and get to it, that won't matter. That's when you risk losing your visitors.
If it's wrong, why is it still around?
The 3-click rule is still around because when it comes down to it, it can help in creating a good website.
While making all of your content accessible in just three clicks isn't some scientifically proven necessity, it's not a bad starting point to shoot for. Considering the rule makes you take a look at your site as a whole: how your content is organized, how it will be accessed, etc.
Multiple levels of navigation and information aren't a horrible thing as long as the paths to those levels are given proper consideration and are clearly marked.
There are similarities to a large menu at a restaurant. I don't tend to like large menus because I get distracted and can never make up my mind but personal preference aside, there's still a wrong way and a right way to approach it.
If the menu items are listed in no particular order and I'm forced to give each and every one equal attention before I'm able to make an informed decision on what I want to eat, I'm likely to feel overwhelmed and unhappy with the whole experience. If the items are categorized well (starters, sandwiches, soups, etc.) then I have a good idea of where I need to begin.
If I sit down at a restaurant and am interested in sharing an appetizer with my friends, I'm going to be looking for the certain cues to lead me to those items. Once there, I expect to be met with options that are actually appetizers. (Filet Mignon with a side of mashed potatoes? That's an entree. You'd undoubtedly do a double take if you saw it listed as an appetizer.)
Your site should work relatively the same. The amount of clicks needed to access something doesn't matter as much as the clarity with which it's organized and the ease with which someone can get on (and stay on) the right path to finding it.