(Sorry if this is obvious or is a duplicate. I couldn't find one.)
Suppose two researchers are studying whether average height of some population has changed significantly. Researcher 1 hypothesizes that there has been some change (two-tailed) and researcher 2 hypothesizes that it has increased (one-tailed). Both want a p-value < 0.05 (as seems to be common in publications).
Researcher 2 will have a lower critical value, so she may be able to reject the null hypothesis when researcher 1 cannot. So now we have one study showing that heights have significantly increased, and another (using the same data and p-value) showing that they haven't changed.
Is that weird? Am I thinking about it wrong? Did I flub something? Even if it's totally correct, wouldn't it lead to misunderstandings? "Studies [on the same data] show that heights have not changed, but have gone up."