While the simplistic answer to the headline is four bits, it’s actually quite a loaded question. A four-bit increase in the scope’s resolution would produce a theoretical improvement of 16 times in the scope’s amplitude dynamic range, from 256:1 to 4096:1. This is quite a significant increase, but being engineers we know that there’s no such thing as a “free lunch.”
Perhaps the first of several questions to ask is how did those four bits get added to the scope’s resolution? The extra bits can be implemented using hardware or software. Most modern, mid-range oscilloscopes include a high-resolution mode, which increases the number of bits by filtering. In essence, bandwidth is traded off for the increased dynamic range. Basically you get one additional bit for each halving of the scope’s bandwidth.
Therefore, to achieve a 4-bit increase, it would reduce the scope’s bandwidth by 16:1. So if you start with a 1-GHz scope, increasing the resolution to 12 bits with the high-resolution mode would reduce the scope’s bandwidth to below 100 MHz.
Alternatively, the scope’s hardware can be improved. The obvious step is to change the 8-bit analog-to-digital converter (ADC) to a 12-bit ADC. This has to be done with some care in order to realize the full benefit of the higher-resolution ADC. Since the 12-bit ADC will enhance the instrument’s dynamic range, the scope’s front end should also be improved to reduce noise; otherwise, the increased resolution is wasted.