Why yes indeed. Although it's more an issue of the ratio of parameters to data points. Consider, for example: if you had 20 data points and used 20 parameters to fit them, you would generate a model that fits all 20 parameters perfectly! But the model would probably be useless for any new data points, because it takes into account all the irrelevant aspects of the 20 data points (their "randomness").

Have a look at: http://en.wikipedia.org/wiki/Overfitting

It is probably a case of overfitting, but it could also be the case that they are fitting an existing model that really has 17 parameters. Without more context, it's a little hard to judge. If I remember, I'll look at the full paper tomorrow. It might make a nice example.

If the model really does have 17 parameters, though, then they need a heck of a lot more than 20 data points to demonstrate its correctness.

Probably, but all I see is a graph and the abstract. It also depends on how the model is being used in the paper. Unless I'm mistaken, Nature puts papers through peer review, which for me personally, anyways, means I'd want to actually read the whole paper before fire and pitchforks.