While it was initially thought that attention was space-based, more recent work has shown that attention can also be object-based, in that observers find it easier to attend to different parts of the same object than to different parts of different objects. Such studies have shown that attention more easily spreads throughout an object than between objects. However, it is not known to what extent attention can be confined to just part of an object and to what extent attending to part of an object necessarily causes the entire object to be attended. We have investigated this question in the context of the multiple object tracking paradigm in which subjects are shown a scene containing a number of identical moving objects and asked to mentally track a subset of them, the targets, while not tracking the remainder, the distractors. Previous work has shown that joining each target to a distractor by a solid connector so that each target-distractor pair forms a single physical object, a technique known as target-distractor merging, makes it hard to track the targets, suggesting that attention cannot be restricted to just parts of objects. However, in that study the target-distractor pairs continuously changed length, which in itself would have made tracking difficult. Here we show that it remains difficult to track the targets even when the target-distractor pairs do not change length and even when the targets can be differentiated from the connectors that join them to the distractors. Our experiments suggest that it is hard to confine attention to just parts of objects, at least in the case of moving objects.