One place that never, ever is full (even during the busy bug fest every summer) is the Science Museum (or NC Museum of Natural Sciences). With a focus on North Carolina, this nice and large museum features a lot of local wildlife and habitats instead of exotic places and animals. Local includes a nice dinosaur exhibit, and also a room with live butterflies. A section on the third floor is reserved for revolving exhibits, right now about the Titanic. Since earlier this year, the museum has a second floor connection across Salisbury Street to the newly built Nature Research Center, deserving of a visit (perhaps with kids 4+) in its own right. The big globe, which is very prominent from the outside features a state-of-the-art multimedia theater inside.
The expansive place is not an indoor playground, but the entire complex is big enough that an inquisitive toddler will walk many miles checking out everything.
This awesome photo is Pterosaurs Ceiling by Stephanie Wallace Photography, used under the CC BY-NC-ND license