There are many signs that a car can give that would indicate oil cooler failure. Unfortunately, these symptoms may also indicate other problems so a visual inspection is often needed of the engine to make sure that the oil cooler is the culprit. The most obvious sign of oil cooler failure is physical distension of the radiator and its associated piping.
If the oil cooler is failing in your vehicle, the first thing you'll notice is decreased performance from the engine. This shows up in the form of slower acceleration and a lower possible top speed. The engine will also register a higher temperature on its heat gauge while running. This is because the engine is not being cooled fast enough and is overheating.
Smoke is another prominent sign as damage to the cooler may actually cause oil to leak out of the engine and into the combustion chambers. This will cause a thick, black, noxious smoke to come out of the exhausts. This smoke is poisonous, and you should turn your car off once it is safe to do so. Further driving the car may irreparably damage the internal parts of the engine and jam it permanently.
Vibration is another symptom of oil cooler failure, especially if oil is getting into the combustion chambers. This will cause mismatched explosions in the various cylinders, which will make the engine vibrate out of sync with its proper firing rate. The vibrations can cause damage to other parts of the car if they are allowed to go on for too long. There may or may not be increased smoke associated with this symptom.
A distended radiator is the only sure sign of damage to the oil cooler, along with broken hoses which would prevent the engine from running in the first place. If a radiator has been stressed beyond factory specifications, this can cause the structure to bend and warp. The radiator will look like a football. In the worst case scenario, this will cause the fins to burst out and look like teeth coming out of the face of the radiator.