It's oft said that we have no evidence of mass volleys of arrows in ancient battles. However, the Roman testudo formation was designed specifically to repel mass volleys of missiles. How do historians reconcile these seemingly contradictory facts?

First, what do you mean by "mass volleys"? Certainly, at times in ancient battles, large numbers of archers and/or crossbowmen shot at mass targets. Medieval evidence for this is given by u/Hergrim in the discussion in https://www.reddit.com/r/AskHistorians/comments/72l2ju/did_archers_not_generally_use_indirect_volley_fire/ and there is ancient evidence, too, e.g., Arrian's Array against the Alans recommends the use of mass archery as soon as the enemy moves into range.

However, if you mean synchronised shooting, where either the entire group of archers, or a selected part of that group, shoots at the same time, we have no ancient evidence. This kind of volley fire became famous with the musket, but it predates it. It may well have been used by ancient armies, especially by ancient Chinese crossbowmen, but the earliest evidence we have for it is Medieval (Tang Chinese). The history of this type of volley fire is discussed in Andrade, Tonio (2016), The Gunpowder Age: China, Military Innovation, and the Rise of the West in World History, Princeton University Press.

While there is a lack of evidence of this kind of volley fire in ancient battles, this doesn't mean that it wasn't used. But whether or not such volley fire was used in ancient battles, it can still be said that we have no evidence.

Second, the testudo was designed to protect against arrows and other missiles. Is there evidence that it was designed specifically to repel mass volleys? Certainly, we see that it was used in other circumstances, such as sieges:

https://en.wikipedia.org/wiki/File:Roman_turtle_formation_on_trajan_column.jpg

Given the general usefulness of the testudo against missiles, including harrassment by archers shooting small numbers of arrows, shooting from above (from walls) in sieges, and many more cases other than mass volleys, the use of the testudo cannot be taken as evidence for the existence of mass volleys.

Thus, there is no contradiction in facts, especially if we are careful to be precise and say that there is no evidence of ancient volley fire (of the second type above) rather than saying that ancient armies didn't use volley fire.