China had some alledgedly under 16 gymnast in their team, which is not allowed and stuff like that. Britain had a 14 year-old diver on their team and no-one is complaing about that.
Then they lied. There is no single age limit for the entire Olympics. The 16 year old age limit is specific to gymnastics. Other sports have different limits (for example, the diving age limit is 14. That's why the teenager from Britain was able to compete).
I thought a rule was if you reach that age in the year of the olympics then you're eligible. For example if your 15 and you're birthday is on the 31st of December then you could still enter the Olympics.
Bobsled (minimum of 14 years old) - Boxing (17-32) - Diving (minimum of 14) - Equestrian (16 or older) - Figure skating (15 by July 1 of previous year) - Gymnastics (must be 16 years old in Olympic year) - Judo (15 or older), luge (16 or older) - Soccer (under 23) - Team handball (over 18) - Weightlifting (17 or older) - Wrestling (must be at least 17 on day of Opening Ceremonies). Source- usolympicteam