No such luck. In TNG and beyond, every single member of the crew went to the Academy! No more basic training in the 24th century. Chief petty officer Miles O'Brien, for example, went through Starfleet Academy, but chief petty officer is an enlisted rating, not a commissioned officer's rank. Nobody thought to mention basic training in the original series, and the brain bug that resulted from that omission led to a ridiculous state of affairs in TNG and beyond, where the entire crew of the ship apparently had to go through the Academy.
However, in TNG "The Drumhead"
PICARD: Please sit down, Mister Tarses. For the record, will you state your name and position?
TARSES: Simon Tarses, Crewman First Class, medical technician.
So he's a crewman. Later:
TARSES: Yes, sir... All my life I wanted to be in Starfleet. I went to the Academy's program for enlisted personnel... took training as a medical technician... served at several outposts... and the day I was posted to the Enterprise was the happiest day of my life.
PICARD: Did you ever consider applying to the Academy... going the whole route and becoming an officer?
[snip stupid sitting under a tree thinking dialouge]
TARSES: No... I was eighteen, and eager... the last thing I wanted to do was spend four years sitting in classrooms... I wanted to be out there... traveling the stars... I didn't want to wait for anything...