I'll focus on second order differential equation.
Suppose the solution for the differential equation is: $$Y(x)=C_1e^{ax}+C_2e^{bx}$$ for this solution we need two boundary conditions and it can be any combination of Dirichlet or Neumann.
But if the solution is $$Y(x)=C_1x+C_2$$ than we can't have two Neumann boundary conditions.
So I suppose the answer to my question is, yes it is matters, the type of boundary conditions.
So the real question is, are there some general guidelines for choosing a-priori the type of boundary conditions? In other words, How can I know that I cant use two Neumann boundary conditions.
The kind of boundary conditions is not dependant from the kind of differential equation.
The boundary conditions are complementary specifications which are added to the given differential equation.
Without boundary condition, the differential equation has a general (or "total") solution including arbitrary parameters. This means that they are an infinity of particular solutions, each one corresponding to a particular set of values of the parameters.
If some boundary conditions are specified, we have to look if a particular solution (among the infinity of them) or several, satisfy to the specified boundary condition. One have to determine if a unique solution exists or several solutions, or no solution.
Of course, it happens that no particular solution satisfies the given boundary conditions. In such a case, the problem, i.e.: differential equation plus not well posed boundary condition(s) , has no solution , even if the differential equation alone has some solutions.