Number of independent components of the electromagnetic field

I'm not really sure what this question is asking that is different from what the other two asked, but possibly the following is what you want. Consider the following, simpler example. In Newtonian mechanics, the gravitational field is the gradient of the potential. So at a point, there are 3 degrees of freedom, because the field is a vector. But if you want to choose how the field varies in a neighborhood, you can't make the variation be however you like. E.g., you can't give it a nonzero curl. Therefore the 3 d.f. at nearby points are in some sense not independent of each other.

When you get the E and B components from the four potential you take the gradient of phi and time derivative of A and the curl of A. Linear dependence/independence may not be preserved across derivatives. This might be the reason why you get additional linear independences in the EM field.

Example: Consider the function f(x) = x^2. Now take a derivative of f w.r.t. x and x^2. You get 2x and 1 respectively. Before the derivative you had one linearly independent component, but after the derivative you have two.