I've had two sites where I've seen this. I have no doubt it exists, but possibly only for some sites/sectors?
The major one was in the answer/tips sector. Flat line at 6000 visits per day for a year. Nothing we did moved this at all
Another in the books side, when we came out of the sandbox for two major terms after 6 months, we gained that traffic but lost an equal amount of long tail traffic, leaving us with exactly the same amount of traffic per day