I'm not sure which formation you're refering to. The romans used many formations depending on the era, the enemy, the grounds and a host of other factors. When you say "Broken line" it could be either the pre-battle quincunx formation or maybe even the Zama formation, both formations are technically "broken line" though the quincunx formation is not an actual battle formation and the zama formation was a very risky move for specific purposes.

Other than these cases romans generally held a solid line, usually in triplex acies (tripple line, most famously in the republic era).

1600- hear me out. It is one thing to ask when the Romans stopped using a broken line, but the formation itself remained effective for quite some time (although perhaps it fell out of usage for a while).

The Spanish army throughout the 16th century adopted many Roman military customs, one of which were the fluid tactical formation called tercios, made up of pikemen, swordsmen, and arquebusiers. Although their base units, the company, was about three times the size of a Roman century, they did utilize a broken line. Reserve units could be brought up to fill in the gaps, or small units of arquebusiers would flank around the enemy unit to provide enfilading fire.

As the so-called Military Revolution of the Sixteenth Century (see book of that title by Geoffrey Parker) progressed, a greater proportion of European army ended up armed with firearms. In 1503, the Spanish army had perhaps 1/6 of their army equipped with guns, by the Battle of the Dunes in 1600, the Dutch fielded an army almost entirely out of units of firearm equipped soldiers. Although it was a very close battle, the Dutch and British defeated a veteran Spanish force, in large part due to superiority of firepower.

Eventually, firepower, and the most efficient formation for its application, a relatively unbroken line, eclipsed the broken lines which had proven so effective for close combat.

If you could specify/diagram which formation you are talking about, that would be great.

In the meantime, if you're up to some light reading, Lendon's Soldiers and Ghosts is a great walkthrough of changes in the Roman army, from the phalanx all the way to Late Antiquity, so you could find your info in there. There is also a well illustrated book, Warfare in the Classical World, which may aid your search.