Experts have been investigating the illness since 2012. They are currently looking into as many as 25 additional cases.
In each case examined by researchers at the University of California, San Francisco, the child suffered paralysis of one or more arms or legs. Doctors say they're not sure if the disease is caused by a virus or something else. All of the victims had the polio vaccine.
Sofia Jarvis of Berkeley lost the use of her left arm after coming down with asthma-like symptoms.
The 4-year-old's case is considered mild. Some of the other children, ages 2 to 16, are paralyzed in more than one limb.
Two years ago, Sophia's parents took her to the hospital when she started wheezing.
Doctors thought it was asthma - but something wasn't quite right. After treatment from her pediatrician didn't help, Sophia spent four days in the hospital. Still, her breathing was not completely clear.
Her doctor suspected it might be pneumonia and gave her an antibiotic, but as the family was leaving the doctor's office the little girl reached her left hand out for a toy, and, her mother said, "mid-grasp her left hand dropped."
Within days, Sophia's arm was paralyzed.
"The prognosis that we've seen so far is not good. Most of the children we've seen have not recovered use of their arm or leg," said Dr. Keith Van Haren with Lucile Packard Children's Hospital.
The average age of children affected by this disease is 12 years old. Researchers want doctors and parents to be on the lookout for signs of the illness, which may begin with cold-like symptoms and include the sudden onset of weakness in the limbs.
Doctors do not know the cause, but they say it's possible that a polio-like virus found in Asia and Australia could have made its way to the U.S.
The Centers for Disease Control and Prevention says it is aware of the findings and caution that most polio-like viruses do not result in paralysis.
UCSF researchers said that the illness is still extremely rare, and not in any way a widespread outbreak, but that they have seen more cases recently than they have seen in the past. They hope to get the word out to physicians so that if they see similar cases they will alert the California Department of Public Health.
ABC News and KGO-TV contributed to this report.