No. A baby’s immune system can and does easily cope with the MMR and pneumococcal vaccines at the same time.
From birth, a baby’s immune systems protect them from the germs that surround them. Without this protection, babies wouldn't be able to cope with the tens of thousands of bacteria and viruses that cover our skin, nose, throat and intestines at all times. This protection carries on throughout life. In theory, a baby could respond effectively to around 10,000 vaccines at any time.