The Tora instructs us to stick to the truth and stay away from
lies. In some exceptional cases, however, the Rabbis acknowledged that
it was permitted or even recommended to withhold the true facts (see this).
The Rabbis also considered that when a person is faced with an
embarrassing situation, he is also allowed to change the story.
For example, if a person goes to see a doctor, a therapist or visits the hospital, and he meets a friend or a neighbor, and he does not wish others to know about his problem, he is allowed to withhold the truth and conceal the real reason for his visiting to the hospital.
It is important to remember that all cases in which we are allowed to withhold the truth are situations where our narrative does not affect, damages or hurts somebody else. If there are two partners in a business, for example, and one of them made an embarrassing mistake that somehow affects the company, he can not conceal his mistake from his partner adducing embarrassment.
Also, a son or a daughter should not lie to his or her parents (or teachers) when they made a mistake, in an attempt to avoid embarrassment. Parents are supposed to know (and accept!) that a son or a student might fail or make a mistake. And adults are there to guide and teach their children on how to learn from those mistakes. By the way, it is very important is for parents to praise our children when, despite the embarrassment, they have the courage to tell us the truth. When children find a mature, confident and empathetic parent who helps the child processing his embarrassing experience in a positive way, that child will have no reason to hide his future mistakes from his parents and will be in a better position to avoid repeating them in the future.
For example, if a person goes to see a doctor, a therapist or visits the hospital, and he meets a friend or a neighbor, and he does not wish others to know about his problem, he is allowed to withhold the truth and conceal the real reason for his visiting to the hospital.
It is important to remember that all cases in which we are allowed to withhold the truth are situations where our narrative does not affect, damages or hurts somebody else. If there are two partners in a business, for example, and one of them made an embarrassing mistake that somehow affects the company, he can not conceal his mistake from his partner adducing embarrassment.
Also, a son or a daughter should not lie to his or her parents (or teachers) when they made a mistake, in an attempt to avoid embarrassment. Parents are supposed to know (and accept!) that a son or a student might fail or make a mistake. And adults are there to guide and teach their children on how to learn from those mistakes. By the way, it is very important is for parents to praise our children when, despite the embarrassment, they have the courage to tell us the truth. When children find a mature, confident and empathetic parent who helps the child processing his embarrassing experience in a positive way, that child will have no reason to hide his future mistakes from his parents and will be in a better position to avoid repeating them in the future.