It is revealed that Jesse is a high school dropout. But in previous seasons, he talks about not wearing his cap at graduation because it would ruin his hair, and he even talks about graduation with his friends at his ten-year high school reunion.
It would be impossible for Danny and Joey (and, in effect, Becky) not to have known that Jesse never graduated from high school. According to Danny at one point in the show, he and Joey have known Jesse since he was at least 13 years old.
The writers did develop a story where Jesse is one day away from inevitably graduating from high school. This happens in episode 09 from Season 4, where Jesse is talking to his high school sweetheart in their "send-off party" one day before; where he tells her that he is not getting in trouble for riding his bike indoors because he is about to graduate.