They will tell you no, but they told me that I do not ovulate every month and I never missed a period unless I was pregnant. I think it might be the "norm" but everyone's body does work differently, to spite what they tell you.
YES, you can. It's called an anovulatory period. If your cycles are highly irregular and very long, they may well be anovulatory. The uterine lining can only get so thick before it MUST be shed whether or not an egg has passed.