In my opinion and the concept of Imamat by Tusi, Imams are born Imams. However they become known to the world only after their appointment by the previous Imam.
Below are two anecdotes quoted form Mumtaz Tajdin's 225 anecdotes which can be accessed at:
http://ismaili.net/source/books/225anecdotes.pdf about the childhood of MSMS worthy of reflection. Both of them involve childhood experiences before he became Imam.
(8) When Imam Sultan Muhammad Shah was yet in childhood, his mother took his proper care of nourishment. Sometimes, she angered and loved him too. In his childhood, the Imam liked sour lemon. Once in the winter, the Imam suffered from cold fever, to which the doctor was called. The doctor checked his pulse and gave medicine and told to Lady Aly Shah, “Don’t allow prince to go outside and also don’t let him to eat sour items, otherwise, it will cause him pneumonia at once.” The mother ordered her son not to leave his bed, otherwise she would punish him. She affectionately kissed him and ordered the house-servant to prepare coffee for him. After drinking coffee, the Imam slept till evening. In the meantime, she came in the kitchen and saw that her son was sucking lemons. She exclaimed, “Muhammad!” Hearing voice of mother, he stood at once and cleaned his mouth and tried to flee out, but she grasped his hand and said, “What are you doing here? The doctor told you not to leave the bed and don’t touch sour items. In this shivering winter, you are sucking juice of lemon.” The face of his mother
became red in anger and said, “Tell me! Why you are mum?” The Imam was frightened and tears burst out of his eyes. She said, “I don’t leave you and will make you forget sucking lemon.” She raised her hand to slap, but one servant came and said, “No, no mother. Don’t beat the prince.” The mother said, “Who are you to interfere? I will also punish you.” He replied, “Don’t worry. You punish me, but don’t punish the prince.” She said, “Why you gave him lemon?” He said, “I have not given. I don’t know when the prince came in the kitchen. I have just seen here before a moment. Nevertheless, I take
his sin on my shoulder. Don’t do anything with the prince.” She said, “It means you favour him.” He said, “Bye-God, I don’t know.” She held hand of her son and went on first floor, where she laid him on the bed and covered him with a blanket and said, “I will lock the room. You stay here tonight. I will now see how you go in the kitchen.” The Imam said, “Mom, I will never go in kitchen, never at all.” But she locked the room and kept its key with her.
Due to the anger of the mother, all the servants were terrified that the prince had been imprisoned. On other side, she was performing worship in a separate room, but could not gain concentration and figuring in mind, “My poor son is alone in the room, which is locked. No servant can enter to know his condition. How is my son? What will be his condition? Leave him alone in the room, which will not be unlocked tonight.” Thus, she tried to revert to her worship. She didn’t dine at night, so the servants too. In midnight,
she told to the servant to bring coffee for her. Suddenly, she saw at the main gate of the bungalow, which itself opened. The light at gate became off. She was surprised to see her son entering the gate, playing with his stick in two fingers. She exclaimed loudly, “Muhammad.” The servants awaked with her voice. His son replied, “Yes, Mom.” He climbed on the stair and came before the mother and tied his both hands round the waist from back side, embraced, and said, “Mom, I have come.” She said, “But I have
locked your room. Who opened it, how you came out? Where you have gone and returned in this shivering winter?”
Hearing it, Imam Sultan Muhammad Shah said,
“O’ poor Mom! I am an Imam in the same way as my father.” Then he continued, “Everything is in the Manifest Imam.”
(9) In 1882, Imam Aga Ali Shah visited Rajkot in India with Lady Aly Shah and his son and successor, Imam Sultan Muhammad Shah. The Imam graced didar to the jamat and said, “Look at my son Sultan Muhammad. In case he may feel loneliness, then you send your sons to play with him. There was one person, called Jivabhai, who had five sons: Devji, Lalji, Somji, Kassim and Nur Muhammad. These and other boys daily played with Imam Sultan Muhammad Shah at evening. One day, Nur Muhammad fell down by a jerk of the Imam. He sustained severe pain and wept too much. The Imam
caught his hand and took him to a safe side. The Imam said, “You have a little injury and crying too much. Keep quiet, otherwise I will hit with hand.” Nur Muhammad became silent. Since he has severe pain, he once again began to weep. This time, the Imam calmly said, “If you don’t complain to my mother, I will give you high tidings of the paradise and make hell unlawful for you.” Nur Muhammad said, “I will not complain to anyone.” The Imam said, “Are you sure that you will not tell to my mother and father. Is it your gentleman promise?” He replied, “Not at all. Then the Imam put his blessed hand on his leg and his pain vanished in a moment.