You can get closer than this, but it depends on the shape of your sail and how large it is. The physics is not intuitive, but the sail is effectively a wing when pointing close to the wind, and it acts exactly like the wing on an airplane does to generate lift. The sail then deflects the wind roughly perpendicular to the chord-line of the sail, and by Newton's second law the lift is then opposite the direction the wind is deflected. As you get closer and closer to 0 degrees relative to the wind less and less of the force is directed parallel to the direction of motion, so the boat gets less and less efficient.
Look at the sails of America's cup boats. Notice how they have a very high aspect ratio? It's so they can efficiently point close to the wind by generating a lot of lift. Now look at a glider. It also has a high aspect ratio for the same reason. Of course, in a glider (or airplane in general) the lift is keeping the plane in the air, while in a boat the lift is pushing the boat along.
You can get closer than this, but it depends on the shape of your sail and how large it is. The physics is not intuitive, but the sail is effectively a wing when pointing close to the wind, and it acts exactly like the wing on an airplane does to generate lift. The sail then deflects the wind roughly perpendicular to the chord-line of the sail, and by Newton's second law the lift is then opposite the direction the wind is deflected. As you get closer and closer to 0 degrees relative to the wind less and less of the force is directed parallel to the direction of motion, so the boat gets less and less efficient.
Look at the sails of America's cup boats. Notice how they have a very high aspect ratio? It's so they can efficiently point close to the wind by generating a lot of lift. Now look at a glider. It also has a high aspect ratio for the same reason. Of course, in a glider (or airplane in general) the lift is keeping the plane in the air, while in a boat the lift is pushing the boat along.
Edit: it also depends on the shape of your keel.