Yes, you can, if the daycare center failed to prevent shaken baby syndrome from occurring while the baby was at the California daycare - or, if there were signs that the baby was being abused outside the daycare and the daycare failed to report suspected child abuse. Allowing someone to shake a baby is unreasonable care, or negligence. It is important to hire a shaken baby syndrome lawyer immediately, though, in order to make the proper allegations in a timely manner.