More than one claim or cause of action can come out of a single transaction, event, or set of facts such as an apartment eviction. For example, it is possible that an apartment manager might have good grounds to evict a tenant at the same time that the tenant has grounds to sue the apartment manager for negligence. Each party can pursue its claim or suit.
An action for eviction is fundamentally an action for breach of contract. The tenant violated the terms of the lease in some way, most commonly by not paying rent. In turn, that breach provides the grounds for the apartment manager to look terminate the lease and evict the tenant.
A negligence action is what's known as a "tort" action. It comes about when someone injures another through negligence, which is unreasonable carelessness. Injury or damage could be to either people or property. For example, suppose that the property manager damaged or lost a tenant's property during the course of eviction: that would be something the tenant could sue for.
Since the negligence cause of action does not depend on or arise out of the lease or any other contract, the fact that the landlord is bringing an action for eviction does not affect the tenant's ability to sue for negligence. These are separate causes of action and grounds for liability.
(Note: you can also sue for intentional bad acts. That means that if an apartment manager put his hands on a tenant and hurt him or her, that's actionable. Or if a tenant is publicly defamed by the apartment manager making false, negative statements--e.g. "John Doe deals drugs"--that's also actionable.)
If a tenant had a legal claim or cause of action that arose before eviction, the fact that the tenant was evicted does NOT cut off his or her right to sue. Eviction is separate from other causes of action--whether or not a tenant is being evicted literally has NO impact on any valid negligence claims.
So if a tenant was injured because the apartment stairs were badly maintained; or assaulted because the building had inadequate security; or sickened by mold spores because the building did not clean up mold conditions--all of those are things a tenant could possibly sue over.
If you or your family were injured at or by your building, apartment manager, or landlord, you may have a cause of action. An attorney can help you understand if you have a cause of action and what it might be worth.
In addition, while your negligence claim is separate from an eviction action, the attorney can also represent you in the eviction and make sure all your rights--in law and under your lease--are protected.