Correct - it *appears* that it happens. But it doesn't actually happen.

What does happen is that inventory continually changes.

Airline move more seats into specific fare classes (prices go down).
People buy those seats (prices go up).
It passes midnight on the east coast (prices go up for some websites, as their "point-of-sale" is on the east coast and it's now past the 21/14/7/3 day advance purchase requires - even if the customer is on the west coast where it's still 9pm).
It passes midnight in Central (Guess what United's website uses as it's point-of-sale!)
Someone who had been searching for prices with the number of people set to 1 decides to buy, but in doing so changes the number of people to 2 which is what they really wanted (price goes up, as there was only 1 seat left in the cheap fare bucket)

And then there's the specific website "glitches" - Search on united.com for a one-way fare, find the one you want, login, and then search again - the price jumps! Why? Because when you logged in it kept most of the search criteria the same, so you didn't realize that it changed it to a round-trip search. Setting it back to one-way will give the original price.

Of course, I'm not saying that there's not some scam websites out there that bump the price when you actually come to pay, or something like that - but the pricing algorithms used by all of the major airlines is well known and well understood, and with the same fares available from multiple places it would be discovered and proven almost immediately if they were to do something like what is being suggested.