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