Fiber optic bundles are used in narrow-diameter medical and industrial instruments for acquiring images from confined locations. Images transmitted through these bundles contain only one pixel of information per fiber core and fail to capture information from the cladding region between cores. Both factors limit the spatial resolution attainable with fiber bundles. We show here that computational imaging (CI) can be combined with spectral coding to overcome these two fundamental limitations and improve spatial resolution in fiber bundle imaging. By acquiring multiple images of a scene with a high-resolution mask pattern imposed, up to 17 pixels of information can be recovered from each fiber core. A dispersive element at the distal end of the bundle imparts a wavelength-dependent lateral shift on light from the object. This enables light that would otherwise be lost at the inter-fiber cladding to be transmitted through adjacent fiber cores. We experimentally demonstrate this approach using synthetic and real objects. Using CI with spectral coding, object features 5× smaller than individual fiber cores were resolved, whereas conventional imaging could only resolve features at least 1.5× larger than each core. In summary, CI combined with spectral coding provides an approach for overcoming the two fundamental limitations of fiber optic bundle imaging.