Can a back injury you recovered from as a child later make you have extreme back pain as an adult?

Depends. It depends on the severity of the injury and the age at the time. If the injury damaged the growth areas of the spine and lead to deformity or there was damage to the disk or joints, that could cause problems later in life.
Spondylolysis. If you sustained a stress fracture of a lumbar veterbra, known as a spondylolysis, over years it can progress to a slippage of the bone called spondylolisthesis. That condition may require eventual surgical correction.