It's possible, but probably not as glorious as you think it is.
One kind is the type of player who relies on other (richer) gamers who want to pay their way to success. Games that are mainly timesinks, and where some rather pay than spend said time. World of Warcraft was sort of known for 'gold farmers' (google it), who pretty much had an entire industry of people doing all the 'boring stuff' of that game. However, though it made some bucks, it was never a huge amount (after deducing electricity and other costs, the money/hour probably isn't much above minimum wage). It's also somewhat risky, as companies want to avoid this kind of behavior. And with more recent titles actually being more 'pay to win', I'm not sure how successful this entire business practice still is.
The other type is esports. If you look at the prizes of some tournaments, then yes...you really CAN make a lot of money when playing games like league of legends, starcraft 2 or a handful of others. The obvious catch is that you have to be GOOD at the game. And I mean REALLY, REALLY, REALLY GOOD!
Still...The former never payed very well, and the latter is never around upgrading characters. So if that was really the message, then yes: the television show was lying.
EDIT: yes...youtube is obviously another possibility: get a channel that's popular enough, and you'll get money from ad revenue.