Is keeping a wild bird illegal?

In most locations, yes. Check with your location's Department of Wildlife (may be called something else, like "Department of Fish & Game") to see if it's true in your area for the specific type of animal you're interested in.

In some cases, injured or orphaned animals may be temporarily cared for by private citizens, but this usually requires the explicit permission of and supervision by the department to insure that it is not being raised as a pet; the goal in such cases is to eventually release it back into the wild.