Is there a way to check if one or some of multiple lines is broken?
Most companies that manufacture printed circuit boards also have equipment that can automatically perform continuity tests (and other tests) on the board's circuit paths as a quality control check.  These tests are usually performed as an automated process. For example, see:


Flying probe testing (EXAMPLE)
Bed of nails testing
In-circuit testing (ICT)
Manufacturing Defect Analyzer (MDA)
Functional Circuit Tester (FCT)
Automated Test Equipment (ATE)


Board manufacturers usually allow customers to specify additional testing regimes such as continuity tests between adjacent pins on IC packages after the ICs are installed onto the board&mdash;e.g., misalignment of the IC package on the pads resulting in shorted pins, solder bridges resulting in shorted pins, etc.

TL;DR: Time is money. Don't waste your time performing complex and tedious tasks that can be performed by machines with far greater speed and reliability.