Instrumental variables (IV) is one of the first econometric methods most students learn. There are many ways to teach IV, but we usually introduce it using a two-state least squares (2SLS) procedure to help build students' intuition about what's going on.
In practice, there are a variety of practical limitations of the 2SLS estimator. For one, it's biased in small samples. And since applied econometricians live in a world of small sample sizes this can be important in practice. Second, if your instruments aren't sufficiently correlated with your regressors -- a failure of instrument "relevance" -- the small-sample bias can be magnified dramatically. In the extreme case of zero instrument relevance the math completely falls apart and the 2SLS estimator isn't even identified.
Below are some notes summarizing the problem. The first 3 pages show the math for the simple case of one regressor one one instrument. The last 3 pages show the general case for K regressors and M > K instruments:
Small Sample Bias of the 2SLS IV Estimator