For us there are many factors that go into the decision to quit an activity.
The first is probably age and maturity level. If a 4 or 5 year old tries ballet and wants to quit at any time - no problem. My daughter has been in ballet since she was 3 1/2. She'll be 8 this month. When she was 6, she went through a period of time around the end of the year when she didn't want to quit, she just didn't want to go "this week." I figured out that she didn't want to go because she was involved in a particular T.V. program, and didn't want to miss the ending of it. So, we talked about that, and just didn't watch T.V. an hour or so before it was time to leave for class. That was it - she really did want to go, but it was very hard for her to break away from the T.V.
From that experience, I learned to really discuss their reasons for quitting with my children, rather than just saying - Oh, you want to quit? O.K. Earlier this summer, my son was ready to move into a new Aikido class. He had been in the 6 to 9 year old group for a year, and was ready to move into the 9 to 12 year old group. (He'll be 10 next month). He has had classes with the new teacher before - and LOVED her. Yet, on the day he was to start the new class, he decided - adamantly - that he "wanted to quit Aikido." I know my son well, and knew he really didn't want to quit, so I made him a deal. He had to go to two weeks worth of classes first (4 classes), and if he still wanted to quit, then he could quit. He was really angry the whole way to the dojo, so I also told him that part of our deal was that he had to fully participate for the 2 weeks. I *never* worry about the money that I've spent on lessons, or that it will be "wasted" if they quit - but I just knew he wanted to quit only because he was afraid to move into the big kids' class. So, I also told him that if he didn't go and fully participate for the 2 weeks, he would have to pay me the $55 for the month's tuition (I'd actually paid several months in advance, so $55 wouldn't have even come close to covering our expenses). My son is a major penny pincher, so I knew this part of the deal would ensure his participation.
He LOVED the class, and from the moment it started, had a huge grin on his face which stayed there throughout the class. He loves this class better than he liked the 6 yo 9 year old class. And guess what? I knew he would! :LOL
Now, if I didn't know my son so well, hadn't been really paying attention to how he truly felt about Aikido (loves it), and was not in tune with what was really holding him back (anxiety about practicing with the big kids), I probably would have just let him quit, and that would've been that. I am so glad I did what I did, though. I wouldn't necessarily even recommend my tactics to other parents, because I handled it in a way that was specific to my son, his interests, and his temperament.
That's why it's so tricky to have a hard and fast rule or policy about these things. If my daughter or son really disliked a class, found it just wasn't for them, or were being mistreated by a teacher they could drop the class immediately, no problems, and no worries about the money spent. In the above two circumstances, I knew there was something more going on than met the eye when they said they didn't want to go, and I'm glad I didn't stick to a policy that said: "If the child says they want to quit, they just quit."
My (as usual) longwinded 2 cents! :LOL