Especially in medical studies, where you've often got cases of n=40 or similar (even in later stages!), this is a huge issue.
In contrast, just think of the size of n you need in physics to be taken seriously!
The major reason for that is, however, that most people in the medical & biological area are rather lacking a profound mathematical education. There are cases where papers get rejected because they are too mathematical.
There are 2 reasons you want a large sample size: (1) to have enough subjects to expect a reasonably representative random sample (typically ~30+ for social science [1]) and (2) to have sufficient statistical power. There's nothing inherently wrong with n=40 and n=40 from an unbiased sample is better than n=400 from a biased sample.
Physicists are generally looking for very very small effects, hence very high n (higher n = higher power = more sensitive to treatment effects). This doesn't mean lower samples sizes are insufficient for other areas of research.
Anyway, the real issue is over-reliance on convenience sampling.
There is a fair share of ignorance of stats to go around, but in the biological sciences and in medicine, getting a big enough n is often practically difficult or impossible. Sometimes it's funding for the study that is to blame. But biology is messier than we often like to admit to ourselves. Sometimes it's just hard to get the materials. I was in a session just this week that showed a gene therapy cure for two kids who were cured of lysosomal storage disorder. It was incredible. But still an n of 4 I think. The process of growing the cells and transferring the genes is so time consuming and difficult, that's the best they could do. That's an extreme case to be sure, but even more mundane stuff can suffer. For example, if your study requires a muscle biopsy instead of a blood test, your n is going to be lower because muscle biopsies are a tough sell to patients. People, animals, and cells are all a lot more fickle than bits and electrons most of us around here are accustomed to working with.
Mathematical biologist here. There are lots of criticisms of the state of mathematical competence in biology I could offer, but power analysis really isn't one of them-- that is typically the one thing that _is_ taught well, if only because data collection can be so difficult. That is not to say that no clinical trials have ever suffered from badly handled statistics, but the issues are usually subtler than "we forgot to determine how many observations we'd need to detect an effect."
The major reason for that is, however, that most people in the medical & biological area are rather lacking a profound mathematical education. There are cases where papers get rejected because they are too mathematical.