Can a child get hand foot and mouth diseases at the same time?

It's all one disease. Hand-foot-mouth disease is one disease caused by coxsackie virus. It's called hand-foot-mouth because there are sores in the mouth and rash on hands and feet.
Virus infection. "hand, foot, and mouth" disease is actually one disease with such a long name. It is caused by the coxsackie virus and results in flulike symptoms and painful blisters/sores in the mouth, on hands and soles of feet in usually young children. There is no treatment except supoort until illness self resolves. Some complications of coxsackie virus include myocarditis.