I believe that any significant physical effect/discovery should be able to be seen by the naked eye in the experimental (or computational) data and should not require curve fitting.However, the graphs below represent a significant counter example to my view.
This is the data which ultimately led to the 2011 Nobel Prize in Physics.
When I see such graphs I am immediately skeptical. But the authors and the broader community were not and subsequent measurements supported the original claims and parameters deduced from this data. In particular the graph below [taken from a Physics Today article Supernovae, Dark Energy, and the Accelarating Universe by Saul Perlmutter] shows how several independent methods constrain the parameters.
I am curious to hear what others think.