Does a neurological exam does not rule out a possibility of a brain tumor?

Correct: Some (many) brain tumors cause no symptoms and a neurologic examination may be normal.

Brain tumor: Although a thorough neurologic examination can identify the location of a deficit, an MRI with contrast is usually required to confirm the cause of the deficit.

Correct: Some patients have a completely normal neurological examination and yet are shown by scans to have a brain tumor, so this is certainly possible.