A large portion of the evidence for object-based attention comes from experiments using the two-rectangle paradigm introduced by Egly, Driver, and Rafal (1994), in which response times are longer when the two stimulus locations relevant to the task are on separate objects. In the new experiments presented here, response times are longer when the two locations are part of the same object but are separated by a concavity in the object, so that the region directly between the two locations is crossed by the object's boundaries. Response times when the two locations are separated by the concavity are not statistically different from when they are on two separate objects. The results are similar for a two-letter comparison task and for a spatial cuing task. Thus, in these experiments, the response time increase does not reflect the cost of shifting attention from object to object, because it appears when the two locations are on the same object, and it is not increased when they are on different objects. Instead, it seems to reflect the complexity of the region between the two stimulus locations. This finding raises questions about whether data from previous two-rectangle experiments should be attributed to object-based attention.