I boiled a whole chicken, then froze the cut up parts. I then cook different meals but want to know if I can refreeze the leftovers from the dinners.
Yes, you can. You're not facing a food safety issue at all except possibly for violating the "time in the danger zone" rule, and then only if, in fact, the rule has been violated. Look at What do I need to know about temperature and food safety? for more on that. For what (admittedly little) it's worth, I find the strict interpretation of that rule (2 hours, cumulative) overly conservative in a home environment assuming healthy, non-pregnant, adult eaters, but that's not my call to make concerning your food.
What you may face is a quality issue. The texture of your chicken may be negatively effected by refreezing. The smaller your pieces of chicken, and the saucier the dish, the less likely it is that you will notice this negative affect on the chicken.