When the number of measurements, N, is larger than the number of unknown parameters, k, and the measurement errors εi are normally distributed then the excess of information contained in (N − k) measurements is used to make statistical predictions about the unknown parameters. This excess of information is referred to as the degrees of freedom of the regression.
Given this definition, if $N$ increases, the degrees of freedom increase as well, but intuitively that would make the problem more constrained (we have more information per parameter). Why is N-k then called degrees of freedom, and it isn't the other way around e.g. (k-N)?