Physics models that are wrong but useful

 "All models are wrong, but some are useful" is a saying usually attributed to statistician George Box. In physics we are often tempted to create a model that might be correct, but ends up being hopelessly useless. 

For example, the multi-particle Schrodinger equation in principle can give us an exact description of the energy levels of any molecule we would like to study, underlying the field of ab-initio quantum chemistry. But it cannot be solved except for the simplest of molecules. Heuristic approximation schemes which may not rigorously justified are essential to obtain useful predictions for large problems of practical interest. String theory is another example, with some arguing it is not even wrong.

There are many neat examples of models that, while wrong, lead to useful predictions and progress in our understanding:

• The Drude model of electrical conductivity. In the original paper there was a fortuitous cancellation of two big errors yielding agreement with experimental data for the specific heat. Nevertheless, the model remains a very good approximation for the frequency-dependent conductivity of metals.
• Conductivity at low temperatures: Before 1911 there were various predictions for the resistivity of metals cooled to zero temperature: zero, a finite value, and even infinite (argued by Lord Kelvin). Efforts to determine which prediction was correct led to the unexpected discovery of superconductivity.
• The Quantum Hall effect: quantization of the Hall conductivity was originally predicted in the absence of scattering, and thus the quantization was expected to only hold to a finite precision. Effects to measure a finite accuracy of the quantization led to the Nobel Prize-winning experiments.

A good model doesn't need to be 100% correct. A good model needs to give an actionable prediction.