I've seen bartenders stack squeezed lime halves and resqueeze them all together. I think that squeezing the peel would yield very bitter juice, but am I wrong?
Yes, you can squeeze citrus too much.
That bitterness you sometimes taste in citrus fruit comes from limonin, a compound that most people can detect at concentrations as small as a few parts per million. In many citrus fruits the limonin is created once the acids of the juice vesicles interact with LARL, a tasteless substance in the fruit's tissues (Hasegawa, 1991). In the juice of a squeezed lime the reaction from LARL to limonin takes a few hours to complete, so if it is very fresh it will not be as bitter.
Over-squeezing, as in using a pestle, will expose more of the tissue (and therefore LARL) resulting in more bitterness later on. If the peels are in the mix, it will also extract some of the oils (mainly limonene) from the peel, which is very bitter. So bitter, that industrial squeezing machines are designed so that the juice extracts never touch the peel.
During summer, guests often ask for caipirinhas at our house. I like to imagine that it is because I peel my limes, remove most of the white pith including stem, and try to mash the limes just before mixing the drink. This way it requires less sugar allowing those great citrus flavors to stand out.
While the pith of citrus fruits can be very bitter, there is no juice in it. To extract any liquid from pith would require much more pressure than any squeezing-based juicing equipment at a bar is likely to generate. A centrifugal juicer is more likely to produce a bitter flavor if the pith is included with the fruit, as small bits of pith will end up in the juice.