Could a delusion be a delusion if the person knows it's a delusion?

Delusions: Psychotic symptoms involve a loss of touch with reality. Patients with psychosis have an impaired reality sense/experience, meaning that what they see, hear, or believe is erroneous or false. However, psychotic patients can have good reality testing. That is, some can identify that the delusion (fixed false belief) or hallucination is false even though their is impairment in reality sense.