Are only MSBs outputted if bits are restricted?

You cannot easily drop a selective number of 0's, you usually just drop a fixed N bits and do the saturation logic as FvM describes, maybe even adding an overflow bit to let other things know that a number has saturated and not actually come out to max. The big problem would be knowing what power of 2 the MSB represents if you tried that, along with all the extra logic required to count the number of 0s and output the next 16.