This just happened to my wife:
She worked at a company for 7 years. She found another job, which is more accommodating to her needs and decided to take it. She likes the people she works with now, so she told the new company she would start in 2 weeks in order to give her current company the professional 2 week notice, not to leave them high and dry. Well, she got an email that day saying her resignation was accepted immediately, so she is no longer with the company. Now we're stuck missing 2 weeks pay and I currently have no health insurance. So much for doing the right thing.