If the assignments are the source of the problem, you can't just take them as "given" and assume that things are OK.
A simpler example: if I assign people to cohorts based on their age, then the results of my experiment may be "conditionally independent" with respect to (say) eye color, but it probably won't be conditionally independent with respect to income or reading level or pant size. In other words, the assumption of independence is violated with respect to all variables correlated with age.
With bandit optimization, our bucket allocation scheme is correlated with time, which makes it nearly impossible to apply conventional statistical tests on any metric that may also be correlated with time. And in web testing, that's nearly everything.