The idea that concepts are embodied by our motor and sensory systems is popular in current theorizing about cognition. Embodied cognition accounts come in different versions and are often contrasted with a purely symbolic amodal view of cognition. Simulation, or the hypothesis that concepts simulate the sensory and motor experience of real world encounters with instances of those concepts, has been prominent in psychology and cognitive neuroscience. Here, with a focus on spatial thought and language, I review some of the evidence cited in support of simulation versions of embodied cognition accounts. While these data are extremely interesting and many of the experiments are elegant, knowing how to best interpret the results is often far from clear. I point out that a quick acceptance of embodied accounts runs the danger of ignoring alternate hypotheses and not scrutinizing neuroscience data critically. I also review recent work from my lab that raises questions about the nature of sensory motor grounding in spatial thought and language. In my view, the question of whether or not cognition is grounded is more fruitfully replaced by questions about gradations in this grounding. A focus on disembodying cognition, or on graded grounding, opens the way to think about how humans abstract. Within neuroscience, I propose that three functional anatomic axes help frame questions about the graded nature of grounded cognition. First, are questions of laterality differences. Do association cortices in both hemispheres instantiate the same kind of sensory or motor information? Second, are questions about ventral dorsal axes. Do neuronal ensembles along this axis shift from conceptual representations of objects to the relationships between objects? Third, are questions about gradients centripetally from sensory and motor cortices towards and within perisylvian cortices. How does sensory and perceptual information become more language-like and then get transformed into language proper?