Fibers may be intertwined to form structures for electronic devices and other parts. Fibers may be intertwined using computer-controlled braiding, weaving, and knitting equipment. Binder materials may be selectively incorporated into the intertwined fibers. By controlling the properties of the intertwined fibers and the patterns of incorporated binder, structures can be formed that include antenna windows, sound-transparent and sound-blocking structures, structures that have integral rigid and flexible portions, and tubes with seamless forked portions. Fiber-based structures such as these may be used to form cables and other parts of headphones or other electronic device accessories, housings for electronic devices such as housings for portable computers, and other structures.