The answer is yes...sort of.
The yacht on Below Deck isn't actually called M/Y Honor. It's M/Y Cuor di Leone, and it's a rentable luxury yacht. The captain, Lee Rosbach, was legit, as was his engineer. But Bravo hired the eight featured crew members, and the actual crew members for the Cuor di Leone were offboard during Bravo's five-week charter.
Captain Lee was less than thrilled with the hired cast. Although they are qualified to do their jobs (at least to a point), he would have fired them if given the opportunity. Per Lee, quoted from the-triton.com:
“The producers of the show wanted to show the long hours and the stress of yachting, but most of that was caused by the crew being inept,” Rosbach said. “On a 50m charter boat, they were all way over their heads. A couple of them tried really hard and they might have made good entry-level crew. I’m worried people will think this is what happens on a 50m charter yacht.
The show was the idea of Rebecca Taylor, who works in TV and has yacht experience. She noted that while keeping the 'real' crew of the boat would have worked great for television, logistically, it wouldn't have worked. She needed all castmates to be American citizens and go through background checks, so the show needed to be cast as any other show would be.
It makes sense that the show is chartered by Bravo, because I highly doubt random guests would want their super expensive vacations aired on TV. Plus, real guests probably would put up a fight when they're kicked off the boat for alleged drug use...the group in the first episode didn't seem to put up too much of a fight. And that whole situation seemed a little staged, don't you think?
I still really enjoy the show, and it's just about as real as other reality TV - sort of, kind of, but not completely truthful.