I call BS. Why would they hire you at an agreed to price, go through the cost & effort and then try and "get you"? Your old neighbour was either in bed with his hiring clients or not very good at his job.
He recruited for wall street trading firms; based on the finance types I've met over the years in NYC I would 100% believe some did this just because they hated losing. He was just making the point that you should never lie about your salary history, because he can help you if you didn't want to give it but he couldn't help you if you BSed everyone and got caught.
It’s definitely made-up. I worked for a Wall Street trading firm in the securities division for 8 years. People lie about comp as a matter of course. Secondly, many/most senior-ish jobs in those firms come with a guaranteed bonus for the first year or couple of years. If they were to fire the person before that period elapsed they would have to pay out the bonus.
I know of many people who were fired the day their guarantee elapsed- I can’t think of a single person on a guarantee who was fired before then. To put this into context, we had a guy on a guarantee on our prop desk who came in for one week after he joined, put on a (massively winning) trade that got him enough to get his guarantee and then literally didn’t set foot in the office for the rest of the year[1] until the day he came in to collect his bonus and resign/be fired.
[1] And it’s not like he was working from home because people in trading were (for compliance reasons) not allowed to work from home unless there was an emergency like a terrorist incident where the trading floor was closed.