One phenomena that definitely annoys me is dealing with irregular observations. This occurs in contexts were data is passively collected based on when people get medical tests. For example, blood pressure is collected when you visit a medical doctor and this information can be used to assess trends in the population.
Here is the problem: people who have no readings often come from two very distinct groups. One is composed of very healthy people who simply have no need of medical services. The second is comprised of poor compliers who should seek medical care but don't. Obviously, the trajectory of these two groups is very different. And, equally obviously, it's hard to argue that these effects will cancel out in a real population.
Inference can still be done but it makes it hard to rule out subtle issues of bias.