Can you get better than 10 bit A to D resolution from the Arduino platform.

In practice your resolution will be lower, because of the noise and other factors.
You may do more measurements (ie. 1000), sum up all the measurements and then divide the sum by 1000 - that is called averaging. You will get better resolution. The noise helps (dithering).

Another approach is smoothing/filtering (low pass filter). You do measurements in a loop where (for example):

..
new_temp = analogRead(1);
new_temp = 0.98*old_temp + 0.02*new_temp;  // =A*old_temp + B*new_temp,  where A+B=1.0
do something with new_temp;
old_temp = new_temp;
..

That will provide a low pass filter which gives you a better resolution. The A+B=0.98+0.02=1.0, you may use other ratio, the lower the B the longer the time constant of the filter..