Alrighty, as someone who does have problems entering the various stages of sleep, I can say that, yes, unless you get to a certain point, you won't have dreams. I've been to so-called sleep specialists at a number of universities and clinics, and at no time have I entered the REM cycle necessary for "dreaming". This has, of course, lead to interesting side effects like a lowered life-span expectancy (apparently I'll be lucky to make it past 70), but there is proof out there that one does not necessarily dream whilst asleep.
In very simplified terms, there are 5 levels of sleep, the last of which, deep REM sleep, involves dreaming. The majority of people get all of these. There are those of us who do not reach that level of sleep, no matter what circumstances or drugs are involved.
Personally, I'm locked out at Stage 4 sleep for the most part, which is the spot before the actual restful sleep (that includes dreaming), so once I pass out, it seems like nothing except time has passed until I wake. Maybe, just maybe, once every few months I'll get to the point where I'm so physically and mentally exhausted that I enter stage 5 sleep and dream, but for the most part, sleep brings no respite from the real world.
Moral of the story, I guess, is that if you're one of the majority who has no sleep problems, don't take it for granted. You're in a much better position than those of us with issues.