Can a player or substitute be added to a roster once the game has started?

In an amateur match, a team asked the assistant referee if he could add a player’s name on the roster – the official allowed the coach to add another name to the roster. This incident occurred during half-time and after all the players and substitutes were previously checked in before the kick-off had started. Is this legal? Is a team allowed to added a player’s name to its roster during the match and/or at half-time?

USSF answer (August 15, 2011):
The rules of the competition (league, tournament, cup, etc.) dictate what will happen in this situation. We will provide the rules used by FIFA for its competitions, but many competitions use different rules. Our motto: Know your local rules!

The following rules appear in the International Football Association Board’s “Questions and Answers 2006,” the last year in which the Q&A were published. As they have not been restated in the Laws of the Game in any other way, they continue to be valid. The particular points occur in the Q&A under Law 3, questions 21-23:

21. A competition rule states that all players must be named before kickoff.
A team lists only nine players and the match begins. May two other players who arrive after play has started take part?

22. If no substitutes have been named and a player is sent off before play has begun, may the affected team complete the side with a player who subsequently arrives?

23. A team reports the substitutes’ names to the referee before the start of the match, but they arrive after the kick-off. Should the referee admit them?