Can an initial value problem have more than one, but still finitely many solutions? That is, can an IVP have only two or three solutions?
Your question does not seem to be clear but here is what I understood from it. There can always be more that one solution to the Initial value problem. But the solutions should be interpreted accordingly to arrive at the desired answer. Say, you are setting up a differential equation to find the heat flow through a bar or rod. You will do so and solve for the temperature of the bar at any point at a given time. See, I'm specifying so many conditions. So when you incorporate these conditions in your obtained general solution, you will arrive at the unique solution.