An electronic device such as a cover for a portable device or other electronic equipment may have circuitry mounted in a housing. The housing may be formed from layers of material such as fabric and polymer layers. The circuitry of the electronic device may include components mounted on a printed circuit. The components may include movable components such as keys in a keyboard. A fabric layer may overlap the keys. Border regions of the fabric layer that surround each key may be characterized by a stiffness. To ensure that the keys or other movable components in the device exhibit satisfactory stiffness levels, the keys can be tested and selected border regions or other fabric layer portions may be laser ablated or otherwise processed to locally reduce fabric layer stiffness.