Is it a problem if the data points used to fit a regression model are not completely independent? For example, we sample n children over 5 years. And we gather 5n data points of age and height. And we do a regression with these 5n data points, trying to fit a line to predict the height given age. Since for each child we have 5 (not quite independent) heights, is this model a good model?
 A: Full independence is not needed, what is really needed is conditional independence (the residuals are independent after accounting for all the predictors).  Sometimes we can create the model such that it gives us the appropriate conditional independence.  
For example in your example if you include a term (or terms) in the model that indicates each child then the assumption of conditional independence is more likely to be reasonable.
Other (probably better) approaches are to use tools like general estimating equations,  linear mixed effects models, or Bayesian hierarchical regression models which model and adjust for potential correlation in the model (but you still need to apply them appropriately).
Other approaches are to fit regular regression (it is the standard errors and inference that are most affected by lack of independence, the estimates of the coefficients are still good estimates) then use a more robust method for any inference (tests or confidence intervals) such as bootstrapping or permutation tests. 
