Why is the mission editor telling me that "only 63 groups per side supported"?

There could be two reasons.

1. You indeed specified more than 63 groups for one side.
   
2. You placed objects in you mission and defined them as civilian. This, too, will take up precious group slots. Make sure that you leave the SIDE field for these objects blank.

(Thanks to squadmate Uzi for the heads-up!)