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We have a related modern question:

WWI trench tactics using shields?

But if we move back into antiquity, are there accounts of the testudo formation withstanding ballistae bolts? Could it?

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No

The Roman Scutum (I assume this is the shield you are referencing to) has been reported to not stand up to the bows in use by the Parthian's in Plutarch's Life of Cassius, in the following line:

6 Thus many died, and the survivors also were incapacitated for fighting. And when Publius urged them to charge the enemy's mail-clad horsemen, they showed him that their hands were riveted to their shields and their feet nailed through and through to the ground, so that they were helpless either for flight or for self-defence.

It is hard to rationalize that the much larger Ballista would have a harder time of piercing a Testudo formation than that of the much smaller Parthian bows. As there were no sources I could find of Ballistas being used against the Roman Scutum.

Update

Here are the facts on ballista's strength and weights data available from Wikipedia:

  • The common projectile of the ballista could weight 26 kilograms (1 greek talent).
  • The maximum range of a normal ballista was 400 meters, which of course wasn't the effective range. The biggest ballista's range was little over a kilometer.

To compare these values to normal bows, here are these two measurements for a typical 50 pounds bow:

  • The arrow weights 5-9 grains per pounds of bow, which means 250-450 grains (250-450/7000 of lb which is around 16-29 grams) plus body, altogether under 100 grams.
  • The maximum range is around 150-200 meters

These two facts makes simply impossible to imagine anybody to stop these projectiles with a hand held shield without getting damaged. The force of these projectiles were lot more than the roman shield could stop, and based on facts the projectiles would go through the testudo, piercing through the first persons shield and knocking the standing people behind by the first person. Their only hope could be only if it does hit only a shield and then the ground, but in testudo it wasn't very likely since the closed formation.

Source: Plutarch's Life of Cassius

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I think the failure to use the Ballista was that it was seen more as a siege weapon and not one of the field. Constructed at sieges against fortifications. –  John Smith May 26 at 5:05
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The Parthians used a recursive bow, that was stronger than the majority of bows the Romans had come up against earlier when the Testudo formation was first developed. The Testudo was still useful against your average barbarian peasants bow. –  John Smith May 26 at 5:25
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@user1095108 the controversy is in part about how well the recurve bows used at the time worked as compared to modern ones. A modern one has no problem penetrating chain maille armour and wood and leather shields, whether a wood and bone laminate bow as would have been used by the Parthians could have done that is in questions (no such of course survive so their exact construction is not well known). –  jwenting May 26 at 8:01
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@user1095108 by changing tactics, by choosing their battles. E.g. if you elect to fight where cavalry is useless, they're no more problem. You don't use infantry to fight cavalry on the open plains of Italy, you do it in the mountain passes instead. Or you send in agents and spies, diplomats and the like, to corrupt the target society from within. –  jwenting May 26 at 10:29
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edited a bit, I think by adding the raw facts on a ballista + bolt vs bow + arrow makes clear how impossible to stop a ballista bolt for a single human who is unlucky to be hit by it. –  CsBalazsHungary May 27 at 9:08
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