Yes, a car fuel pump can run, but it can still be bad. Here are some points to explain how this can happen:

Inconsistent Fuel Pressure:
A fuel pump might operate but fail to maintain consistent pressure, leading to engine performance issues such as hesitation, stalling, or poor acceleration.

Reduced Flow Rate:
The pump might be running but unable to deliver the required amount of fuel to the engine, causing the vehicle to run lean or not start at all.

Electrical Issues:
Intermittent electrical problems can cause the fuel pump to cut in and out, which may not be immediately noticeable but can lead to unpredictable engine behavior.

Mechanical Wear:
Internal wear and tear, such as worn-out bearings or impellers, can reduce the pump's efficiency even though it is still running, resulting in insufficient fuel supply.

Clogged Fuel Filter:
A clogged fuel filter can cause the pump to work harder, potentially running but failing to provide adequate fuel flow to the engine.

Overheating:
A fuel pump that runs but overheats can lead to intermittent failures. In these, the pump shuts down temporarily and then restarts, causing sporadic engine issues.

Noisy Operation:
A running fuel pump but making unusual noises (such as whining or humming) can be a sign of impending failure or internal damage.

Poor Engine Performance:
The engine may exhibit symptoms like misfires, rough idling, or difficulty starting even though the fuel pump is running, indicating a problem with the pump's ability to deliver fuel properly.

Fuel Contamination:
Contaminants in the fuel can damage the pump or clog the fuel lines, causing the pump to run but fail to function effectively.

Diagnosis Difficulties:
A running fuel pump may pass initial diagnostic tests but still fail under specific conditions, making it challenging to diagnose without a thorough inspection.