How many conditions do we need for a problem to have an unique solution? How do we know how many initial and boundary conditions we need for a problem to have an unique solution ?? 
For example if we have the problem $$u_{tt}-u_{xxtt}(x,t)-u_{xx}(x,t)=f(x,t), 0<x<1, t>0$$ how many conditions do we need?? 
 A: I can't say for sure in the general case, but I can tell you with certainty that in the case of Laplace's equation, the question isn't 'how many boundary conditions do I need' but rather 'how many conditions make up a boundary.' 
Suppose u(x,y) is a solution of Laplace's equation in the plane on some open set S that is bounded by a simple (non-self-intersecting) closed curve. By definition u(x,y) is a harmonic function, and thus there is a holomorphic function f with real part u and an imaginary part that is also harmonic (u's harmonic conjugate). By the Jordan curve theorem, we know that S is simply connected, and then by the Riemann mapping theorem, we know that S is related to the open unit disk in the complex plane by a conformal map g. That means that g * f is a harmonic (implies holomorphic) function on the unit disk, and so has a unique Laurent series there if we know f on the boundary, for which we only need to know u on the boundary. That uniqueness carries backwards to f, since g is bijective (as a requirement of being conformal). And that uniqueness clearly carries back to u(x,y). 
Shorter version: it's not about how many boundary conditions you have. It's a question of whether or not they make up a complete bounding curve of the region you're solving over. If it's a square, you could have four boundary conditions. If it's an n-sided polygon, you'll need n boundary conditions. 
P.S.: This argument does generalize to Laplace's equation 3d, but it just requires piecing together spheres from planes with conformal maps (i.e., the Riemann sphere). The uniqueness of solutions of Poisson's equations on a region bounded by a simple closed curve is clear from playing with Green's functions.
