No, thats why Z-buffering was invented.
The painters algorithm (from what I understand) starts with the furthest away polygon, and drawns every single poly to the screen that is within the clipping area. It overdrws each pixel potentially many, many times - imagine you are looking into a forest, the painters algorithm would draw the far away trees, and progressiely come closer to you, drawing each closer tree over the top of the further ones.
z-buffering, on the otherhand, draws the closest polys first, and then if the further polys are only drawn if there is no poly at that location that is closer. This way each pixel is written to exactly once. Therefore, if a single tree is blocking your whole view, NO trees behind it are drawn.
You may say that drawing pixels to the screen (LCD) doesn't take much time, but each pixel also has texture and lighting calculations associated with it, and if you could test whether the pixel was going to be visible first, you could leave out those texture and lighting calculations, as well as the actual write of the pixel.
of course, correct me if I'm wrong about the painters algorithm (z-buffering I'm pretty sure is right)