Brachial Plexus Injuries can occur at any time in life, but the risk to babies is more likely to occur in the womb or during delivery. So often they are not obvious until the baby is a couple of months old and not moving the arm equal to the other arm, and sometimes not even moving the arm at all. Another symptom I see is that the baby's clavical (collar bone) starts growing more forward due to the nerve not stimulating proper bone growth.
Here is a link describing brachial plexus injuries:
Brachial Plexus Injury in Infants, Health Facts For You, UW Health, University of Wisconsin Hospital, Madison
If you are concerned that your infant does not move one arm as well as the other, please discuss this with your pediatrician. An evaluation with a neurologist may be necessary to rule out a brachial plexus injury. Other problems that it could be include hemiplegic cerebral palsy, torticollis (neck muscle), orthopedic impairments, nerve impingement, and other things that should be looked at by a physician. The baby may need physical and/or occupational therapy to help learn to use the arm better as well as to provide stretches/exercises.