Short answer. It is limited. Article 106 of the Labor Code makes the client jointly and severally liable with your contractor "to the extent of the work performed under the contract." Wages tied to the assignment you actually performed for that client are covered; unrelated debts from your agency, or work done for other clients, are not.
What the law says
the employer shall be jointly and severally liable with his contractor or subcontractor to such employees to the extent of the work performed under the contract, in the same manner and extent that he is liable to employees directly employed by him
Labor Code, Article 106 — Contractor Or Subcontractor. Read the full provision →
The statutory limit: "to the extent of the work performed"
Article 106 provides that if the contractor fails to pay the wages of his employees, the employer shall be jointly and severally liable with his contractor or subcontractor to such employees to the extent of the work performed under the contract. The phrase "to the extent of the work performed under the contract" is the boundary: the client answers for wages that correspond to the work you did for that client under that engagement, not for your entire employment history with your agency.
What this covers in practice
If you were deployed to one client and your agency failed to pay the wages earned during that deployment, the client is jointly and severally liable for those wages — meaning you can pursue either the agency or the client for the full amount owed for that work. The article also says this liability runs in the same manner and extent that he is liable to employees directly employed by him, so it is treated the same as a direct wage obligation, just scoped to that assignment.
What falls outside a single client's liability
If your agency owes you wages from a different assignment, with a different client, or from periods when you were not deployed anywhere, that particular client's liability under Article 106 does not extend there — each client answers for the work performed under its own contract with the agency. Unpaid wages unconnected to any contract, such as during a bench period between postings, are a claim against the agency as your direct employer, not against a client that never engaged you for that period.
Sorting out what you are owed
Because liability is tied to a specific contract and assignment, the practical step is to separate your pay records by which client and which period they cover. Payslips, deployment orders, and time sheets showing where you worked and when let you and, if needed, a lawyer identify which client is liable for which portion of what you are owed, and which portion instead falls on the agency alone.