Not unless you really work at it. While you probably don’t want to stick magnets directly to your computer case, having them nearby will not harm your computer. Magnets can damage floppy disks and magnetic tape storage media, so you must keep magnets away from these items. They will not, however, damage any data on your hard drive unless you place a very large and powerful magnet directly on top of the drive.   Every hard drive already contains a powerful neodymium magnet, so one moving around outside the case will not affect the data.