I had five shots at one time when I joined my ship in the Philippines — medical malpractice. A couple of days later, I developed a sebaceous cyst in the shoulder the size of a baseball. Sick Bay refused to treat it — more medical malpractice. I cut and drained the cyst myself, no anaesthetic. Still bothers me on occasion, 30+ years later. If five are too much for an adult, six almost certainly are too much for a baby or toddler.
I had something similar 30 years ago from a single tetanus shot. A baseball sized lump the doc drained and cultured. He said it was an aseptic abscess. We decided it was from the Thimerosol preservative. I knew I was sensitive to it because I had a reaction to contact lenses solution containing it. I’ve had flu shots for the last 20 years with no problem. I always check to make sure ther is no Thimerosal.