How much can we downsample by? and how much can we upsample by? e.g. let's say we have a signal with B = 100 Hz that is sampled at 400 Hz. The anti-aliasing filter will remove replicas, and there is nothing to alias, and we downsample. If we downsample by 2, we will get signal of 200 Hz. But what's stopping us? Can we downsample by 4, or 8? 

(why can't we downsample by 4, get B = 400Hz, then have a new sampling rate at 800 Hz?)

Similarly, is there a limit to how much we want to upsample?
Nyquist. If your signal is real and has a BW of B = 100 Hz (meaning DC to 100 Hz), then the most you can downsample by is 2, in order to obey Nyquist and have a sampling rate that is at least 2B = 200 Hz.  Note you will typically want to sample more beyond this limit to allow room for reasonable filtering.