To answer the question you asked, certainly a PG kid can be overscheduled. Whether yours is is of course another matter. But it sounds as though you're wanting to explore the theory that lots of organised out-of-school activities can be a generally applicable antidote to an understimulating school environment, and I feel positive that as a general rule that won't work. This is not to deny that it might work for an individual child!
My DS does two or three one-hour afterschool activities at school - and they aren't skill-focused ones, either - and that is absolutely enough for him, to such an extent that weekends when he has two parties are really too much. For him, I think what he gets too much of is organised group activities, of which school counts as one. He finds it tiring to deal with group interaction, and needs time alone. Whether an individual child feels that way is personality rather than giftedness, but I guess that PG children are more likely to feel that way than others, because they are likely to have additional stress in the interactions imposed by their outlier status.
