There have been a lot of helpful and informative replies to these questions, and I just wanted to supplement those with some Core Values observations.
FLL creates an annual Challenge as a way to spur interest and excitement in teams about robotics, solving problems, and applying technology to achieve goals. Teams only get a limited time to work on the Challenge in a competitive way, and that focuses interest, and introduces some real-world engineering constraints. (This is the way that all FIRST sponsored robotics programs work.)
As it says in the Coaches' Handbook, teams learn that friendly competition and mutual gain are not separate goals. The Core Values include:
We learn together.
What we discover is more important that what we win.
We have FUN!
It is true that FLL tournaments have a competitive element--that adds some spice to the mix. But what I'm looking for as a referee or judge is that teams are competing with their own personal bests--to do the best that they ever have with this Challenge. Our goal in putting on tournaments is to allow the teams to "show what they can do"; they've worked all season long, and this is our opportunity to see that and celebrate what they have accomplished.
There is, officially, no such thing as losing an FLL tournament. There's a participation award for every child (and often coaches), and most of us long-time volunteers recognize that this is the most important award that we give out. The experiences that the team has are the real long-term reward for the work they put in during the season. (And many coaches have heard from me that the most important part of the tournament is not when the teams meet with the judges and referees...)
Yes, we do have awards to recognize excellence, and to provide an additional incentive and inspiration for all teams to do their best, and the hierarchy of tournaments is part of that structure. But you'll see that the journey really is its own reward.
So, if your team goes to a tournament and is not invited to advance to the next level, is that all there is--are they done? Not necessarily.
There's probably a break, to relax and get over any disappointment. But after tournaments, team members are often full of new ideas, that they are eager to tinker with and try out. We hope that they continue to improve their skills with robotics, and problem solving, and sharing their experiences with others. We encourage teams in areas with multiple tournaments to attend one before or after their tournament, just to see more ideas, and stay excited. There are also opportunities to attend via the internet, locally and around the world, if you aren't able to go to one in person, and to be in the audience for FTC, FRC or other robotics events. Or mentor a Junior FLL team!
As Nick Swayne said, there is some interest in friendly events in the springtime; I know that there are some opportunities already, and teams can organize and host a friendly get-together with another group.