Can calibration improve the accuracy of a circuit?
Keep in mind that anything adjustable will eventually require (re)adjusting. 

The actual accuracy of a complete circuit has to take into account tolerance buildup; all the part tolerances have to be assessed. Certainly, an adjustable part (potentiometer in your case) can increase the accuracy of a circuit, at the expense of needing to go through the same procedure regularly.

Your 1% resistors will have an initial resistance of 1%, but over time and temperature, this is more likely to be of the order of 3% (I work in an industry where we support equipment for decades so such an analysis is important for us).

Sometimes it is worth adding circuitry to add an Automated Calibration feature that could perhaps invoke at power-up or on some regular basis, although that would be bit over the top for a simple resistive network. This is useful for high end test equipment, though.

The simple answer - yes, you can get better than 1% (provided your measurement equipment is better than 1%). The general rule of thumb is that to achieve a particular accuracy, the measurement equipment should be 10 times as accurate(but this varies depending on who you ask).