Before I was born, my family were members of a denomination that practiced infant baptism. My two sisters and my brother were all baptized as babies. This was no mere 'infant dedication' but rather as francisdesales mentioned, the replacement for circumcision; the sign and seal of God for His people.
My family fell away from faith but later came back. This is when the infant baptisms of my siblings came into play. My brother became part of a Baptist congregation that baptizes people 'into the church'. So, everyone who became a member of that fellowship was baptized no matter what. When my brother and s-i-l moved to a different state, they were baptized yet again. I think my brother has been baptized four different times now. I (as well as a few other members of my family) had several conversations with him about the fact that this is a false practice and he refuses to be baptized ever again.
My sisters started going to the same fellowship I attended. Now, my eldest sister, upon her regeneration, considered her infant baptism. She talked with my mom and dad, and learned of the vows that were spoken, and the reasons why they had baptized her as an infant. She looked at her regeneration as the fulfillment of God's promise that baptism is His sign and seal, 'a circumcision made without hands', and that He had been faithful to His promise to be 'raised up with Him through faith in the working of God" . So, she finds her infant baptism as effectual as a symbol of her death in Christ and her life in Christ. With the consideration that there is but ONE baptism, she was not rebaptised.
My other sister though looked at her baptism differently. She feels strongly in the believer's baptism. She doesn't disagree that baptism replaces circumcision as God's sign and seal, but just as God changed His sign from circumcision to baptism, so also He changed the 'rules' as it were, about who is to receive this sign. True, the Hebrews were commanded to circumcise their infant boys, as Jesus was circumcised. But, just as Jesus, although being circumcised, was also baptized at an age of understanding, my sister believes that baptism is for believers who understand and agree with the proceeding.
I agree with her. We didn't have our kids as infants, and have no knowledge about whether they were ever baptized. If we had them as infants, we would not have baptized them. Our daughter was baptized soon after her first confession of faith, at the age of 6. At 6 years of age, she believed in God, she understood her own sinfulness, and she believed that Jesus died for that sin and was raised up again. While the belief was basic, it was sincere. My son is 7 now, and understands and believes these things as well. He has made his confession of faith and readily shares his faith with others. The only reason why he isn't baptized yet is because the fellowship we were going to had only one baptism service a year, and we missed it due to family conflicts. While no date is set, he will be baptized this summer.
I agree that practising infant baptism can create confusion. But, the decision to be baptized again is a serious one and requires prayer, study in the word, and some practical advice from one's pastor and elders probably isn't a bad idea either. It is also a personal decision and whichever way a believer decides to handle their own infant baptism should be treated with respect by others.