> The choice between leaving and being fired may have an impact on your next job, perhaps??
The impact comes not from what they tell you, it's more about what they'll tell the next company when they call to ask about you (though if the two don't match up, you run into a lot of problems).
If the company has documentation to back up their side - something like "he/she was found asleep at his desk several times (dates A, B, C), was given both verbal (on date D) and written (on date E) reprimands, continued to sleep at his/her desk (dates F and G), and was let go on date H" - then I don't believe that there's a law that stops them from doing so.
Yes, they may get sued - but they're likely to win the lawsuit (especially if they required the employee to sign the written warning, so he/she can't claim ignorance).
The impact comes not from what they tell you, it's more about what they'll tell the next company when they call to ask about you (though if the two don't match up, you run into a lot of problems).