For follow-up discussion and general commentary on the topic. Comments are sorted chronologically.
Mark 6:5-6 Now He could do no mighty work there, except that He laid His hands on a few sick people and healed them. And He marvelled because of their unbelief. Then He went about the villages in a circuit, teaching.
These are the only verses that I can think of right off the top of my head that say, at least to me, that Jesus did not heal some infants, perhaps, because of the unbelief of the parents.