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.