Well, there are some advantages and disadvantages for specific tasks. I've examined the case where we try to do some modifications on the bit number of the samples just after the ADC stage.
So, let's say that we feed a periodic square wave, which jumps between, say, 0 V and 5 V, to a 14-bit ADC. Also, say that the noise power on the input signal is so low so that during conversion, amplitude changes on the input signal due to the noise is below the resolution of the ADC in terms of voltage. In this case, 0 V and 5 V are represented as $00000000000000$ and $11111111111111$, respectively. Theoretically, we can throw out those 13 bits on the right hand side of the MSB (most significant bit) of those two binary representations because we just have two levels to show. Therefore, by doing so;
Bit efficiency is increased as now, we represent the two levels with the minimum required amount of bits, i.e., 0 for 0 V and 1 for 5 V.
Amount of time to process each sample is reduced as the upcoming sections of the system deals with comparably small bit count at a given time. Thus, the overall computation time is reduced.
Also, which depends on the designer, complexity of the DSP system may be decreased.
The above outcomes are more likely to occur under ideal and suitable conditions.
So, what if we work with an analog arbitrary signal which has a variety of voltage levels? Again, assume that the noise effects on the input signal is very low.
If we want to illustrate that signal in the discrete domain as close as possible to its analog form, keeping the amount of bits to show each sample is a good choice as, for example; some voltage levels may be gathered and shown as $11111111111111$, $11111111111110$, and $11111111111101$. If the last two or more bits of those binary information is omitted, all of the three different levels of the input signal becomes equal in the discrete form. This, depends on how precise the system should be according to the needs, might cause problems as the difference between the input signal and the truncated version of its sampled version is increased in addition to the analog information loss that happens during the ADC action.
There may be other drawbacks of such a manipulation on the bit amount, especially during error detection and correction in digital communication systems which could be represented by the following basic schematic.
In the figure above; if we remove some of the LSBs (least significant bit) of the binary numbers that are created after the line decoder, we are likely to fail during the error correction phase of the information exchange which may lead to serious problems.