That is exactly what happened to my Coworker. She interviewed for a public job and they gave her the thumbs up. She passed the background check, and gave her two weeks notice.

Then, some inconsistency came up. Apparently, on her application she checked the box for never having smoked weed. But on a second document she checked the "Yes" box. This is California 2015, they asked her to do a polygraph test. She failed.

Couple days later, she called the first job to see if she can come back. They were bitter that she left, but because of an extremely high turnover rate, they let her back in. It was never the same for her.

So yeah, have a guarantee at the new job before you quit the first one.