The only downside - it required a reliable and precise implementation that took a lot of efforts. I always used IIOP.NET for most of my gigs and it was excellent. I also ended up as an active IIOP.NET contributor.
CORBA ORB implementation was a fine art a few could grasp. And this was the biggest drawback - the standard was (and is) excellent, but most implementations tend to be complex and shaky.
I still actively use CORBA. The server usually offers two kind of endpoints: REST for third-party integrations (which are usually naive and simplistic), and CORBA for the system itself. I've built nice things with such architecture that involved worldwide deployments including embedded hardware. I am very proud of my involvement and the fact that I could help to improve the everyday routine for many people worldwide.