"Many hands make light work" - John Heywood.
Along with what Don said, on our team (1676) we also separate ourselves into four divisions: Power and Control, Mechanical, Info Tech, and Non Tech. These are then broken down further into four sub-divisions. These 16 sub-divisions (some teams call them sub-teams instead; same basic idea) each has their own tasks to do, and each specializes in an area that contributes to the whole entity of our team.
For example, one mechanical sub-division works on one main part of the end effector, another mechanical sub-division works on some other main part, the electrical sub-division wires the robot, the carpentry sub-division builds the field and performs other woodworking jobs, and more. Obviously we have others such as programming which programs the robot, pneumatics which manages pneumatics, chassis which creates our chassis, etc.
In addition to our mechanical sub-divisions (Power and Control, and Mechanical), we have two divisions for Non-Tech and Info Tech, which focus on the nonmechanical aspects of our team, such as fundraising/sustainability, CAD, animation, website, awards, bookkeeping, and media. Remember, it's not just about the robot!
We have dedicated team members in charge of each division and sub-division, which ensure that each group stays on task and gets finished on time. The division leaders look after the sub-division leaders, who manage their sub-division. Above the division leaders are two C.E.O.'s (one from each school in our school district), which look over the division leaders, and do other work such as attendance and such.
The key is organization. Obviously not every team divides themselves up the same, but if a team is organized, focuses on what they are doing (and understands what has to be done), and is well managed, they will without a doubt succeed. We now have 100 team members this year compared to 50 in years past, and are able to control the otherwise entropy by making sure that every member has a part on the team and knows what their job is.
I hope this helps! Feel free to ask if you have any more questions =).