Yes, always have done. I get Walla Sweet onions that are much bigger than softballs, and storage onions about softball size.

That's a 20" diameter bowl of Walla Sweets from last September....

I've never used onion sets, so I can't comment. I have worked since 2011 to balance my soil, and every year I see improvement in the vegetables and plants grown there. So all I can say is to get a complete soil test NOT from a University, and start amending the soil. Big onions will follow....