The odds of you eventually coming across dependency in your data is very high. Using standard methods that ignore the situation can lead to problematic inference. More to the point, you would be missing out on a much richer story to tell with the data. One can estimate a variety of cluster-specific effects and incorporate multiple types of structure, all while still be able to talk about global effects as well. One can also extend such models to other types of ‘random effects’ approaches as well.
With the right tools, even complicated mixed models can be fit relatively easily and quickly. It does take some getting used to, but in the end can be a highly satisfying modeling approach. Use them the next time you encounter some dependency in your data.