Vreference and A/D converts

What effect does changing the Voltage reference for a A/D converter have on each bit? Ie, if our reference voltage is 5V, and we have a 10 bit A/D, that means there are 1024 Different levels it can measure, and each bit is 4.88 mV. If we decrease Vref, does this increase Resolution and allow us to measure smaller signals, and if so, at what cost (ie tradeoffs)?

The value of the LSB would decrease by the ratio of the new reference voltage to the original reference voltage. The cost is that the A/D converter has an intrinsic noise level which will likely show up as more bits of noise.