That time was the '60s -- i.e. right out of the gate -- as X-Men sales declined with every issue. By 1970, they were at the very bottom of Marvel's sales chart, so the series was cancelled, and for the next five years it was kept on newsstands only as reprints. In 1975, the series was rebooted with Giant-Size X-Men (referring to the length of the comic, not the physical stature of the heroes), featuring a mostly new cast with more diverse backgrounds and viewpoints, no doubt prompting several complaints about childhoods being ruined. The new X-Men introduced Colossus, Nightcrawler, Storm, and the super popular one that you should definitely know if you want to consider yourself culturally relevant, Thunderbird. The new team also bought in Wolverine, who had recently been introduced in a Hulk comic, of all places, and they recycled Cyclops from the last run, because every superhero team needs at least one member whom everyone hates and hopes dies.
The revamped team underwent a few tweaks, but eventually they caught on and X-Men became one of the best-selling comics ever. Their success was thanks in part to breakout star Wolverine, who was almost called "the Badger" instead. You can't imagine a massive film franchise anchored by Hugh Jackman's Badger, can you?
20th Century Fox