Can you get diarrhea from drinking too much diet soda and coffee?

Yes. Stimulants and artificial sweeteners can cause diarrhea. Excess of either coffee or diet soda should be avoided.
Coffee definitely. Coffee (even decaff) contains organic acids that can be irritating to the lining of the gut, and cause diarrhea. Excess caffeine (even in diet sodas) can as well. Some people are sensitive to fructose as contained in table sugar or high fructose corn syrup (not really hi fructose, but corn syrup is solely glucose). Excess carbonation can stimulate bowels. Artificial sweeteners don't usually cause.