Illustrations of history

Fatigue. It's a tiring position to hold, so as long as you constantly pepper them with arrows or other missiles, eventually, some will get through. Most times enemy troops would run out of ammunition long before that, but there are instances, such as the Battle of Carrhae, where constant resupply of ammunition made the constant fire effective. The Roman commander at Carrhae thought he could 'outwait' the enemy's ammunition, which normally would be a good bet, except that the enemy had arranged for supply trains of camels laden with nothing-but-arrows so that they could endlessly shoot at the Romans.

The other method is that it's a terrible close-range position, since it requires everyone to hold a pretty static stance. This was also exploited at the Battle of Carrhae, with troops in testudo formation charged by heavy cavalry whenever they formed up.

Testudo formations usually cover the sides if there are no allied forces on the flank, or, on rare occasion, even cover the rear (if that was a concern).