Like Frank, (and many other non-cave trained divers) I've been to the grate at Vortex. I was on a "trust me" dive with a much more experienced friend (but also not cave trained). It was probably around dive #20 and looking back, I had no business there. I've also been to Morrison many times without overhead training. If, at the time, you asked about the danger associated with those dives, I would have indicated that I thought it was minimal. Then a few months ago I took a Cavern Course at Ginnie Springs. Besides being an incredible class for your trim and propulsion techniques, the course really opens your eyes to the dangers of overhead environments. Most cavern/cave instructors also instill in their students the importance of warning the untrained about those dangers. To cave divers, any accident or fatality in a cave or cavern is a step closer to having their access to caves limited or completely cut off. From what I gather, most cave divers would rather upset you or down right piss you off than see you get hurt in a place you "shouldn't" be. While some are more diplomatic than others, most (if not all) are not trying to be holier-than-thou. They are just aware of dangers you are probably ignorant to.
@ Frank: If you have the opportunity, take the Cavern Class at Ginnie Springs. Like you, I was much more interested in wrecks and reefs, so I wasn't sure I wanted to continue training past the Cavern level. Then we did our last training dive at Devil's Ear. It makes the cavern/cave at Vortex look pretty plain vanilla. All I could think after the dive was "I've got to get my Intro Cert so I can see what is past the daylight zone in that cave." Even if the class doesn't light your fire for actual cave diving, I'll wager it will be the best class you've taken to date.