I should think you can, depending on the wood, many can be toxic.

The bark of a Willow tree is used to make Aspirin, we smoke paper and eat many plants with less woody stems. There are certain other barks and cambium (the soft layer between the bark and the wood) that contain nutrients, such as birch, pine, elm and a few others that have been eaten by our ancestors for centuries and even have medicinal properties. We also grate cinnamon and a few others as spice. Dog food is often bulked up with ash.

The real issue is that the hard cellulose in the actual wood part is not particularly digestible and basically pure fibre and devoid of any real nutrient value. So it would need to be boiled or blended first I imagine, or steeped as a tea. It would be revolting or taste like nothing and probably give you constipation but I doubt you would die.

As a raw bite of a chunk of wood, no. It would be considered inedible.