No, it is not possible to be schizophrenic yet neuropsychologically normal.

Cognitive impairment is well documented in schizophrenia, though some reports have been interpreted to suggest that it is possible to have schizophrenia without neuropsychological impairment. The authors tested this by comparing the neuropsychological profiles of closely matched patients with schizophrenia and healthy comparison participants. Sixty-four patients with schizophrenia and 64 healthy comparison cases, matched to within 3 Full-Scale IQ points, were tested using the Wechsler Adult Intelligence Scale (3rd ed.; D. Wechsler, 1997b) and the Wechsler Memory Scale (3rd ed.; D. Wechsler, 1997c). Neuropsychological profiles for these groups were markedly different, with the group of patients with schizophrenia exhibiting performance deficits in memory and speeded visual processing but superior verbal comprehension and perceptual organization relative to the group of healthy comparison participants matched on Full-Scale IQ. Thus, scoring in the normal range does not preclude neuropsychological abnormality in schizophrenia, confirming that neuropsychological impairment is a core feature of the illness.