I experienced this, waaaay back in the web 1.0 days. I forget which airline (maybe Southwest because you can't buy their tickets through a third party), but they would offer a pattern of low-high-medium prices in response to multiple searches. I was able to reliably trigger the pattern by clearing cookies.

However, I don't think anyone does this today. There are too many online ticket agent (including ITA), so it becomes obvious when the lowest fare for a given (city-pair, flight number, class of service, airline) tuple isn't being offered.